0,4

Number of linear extensions of the "zig-zag" poset. See ch. 3, prob. 23 of Stanley. - Mitch Harris, Dec 27 2005

Number of increasing 0-1-2 trees on n vertices. - David Callan, Dec 22 2006

Also the number of refinements of partitions. - Heinz-Richard Halder (halder.bichl(AT)t-online.de), Mar 07 2008

The ratio a(n)/n! is also the probability that n numbers x1,x2,..,xn randomly chosen uniformly and independently in [0,1] satisfy x1>x2<x3>x4<...xn. - Pietro Majer, Jul 13 2009

For n >= 2, a(n-2) = number of permutations w of an ordered n-set {x_1 < ... x_n} with the following properties: w(1)=x_n, w(n)=x_{n-1}, w(2)>w(n-1), and neither any subword of w, nor its reversal, has the first three properties. The count is unchanged if the third condition is replaced with w(2)<w(n-1). - Jeremy L. Martin, Mar 26 2010

A partition of zigzag permutations of order n+1 by the smallest or the largest, whichever is behind. This partition has the same recurrent relation as increasing 1-2 trees of order n, by induction the bijection follows. - Wenjin Woan, May 06 2011

As can be seen from the asymptotics given in the FORMULA section, one has 2*n*a(n-1)/a(n) -> Pi as n->oo; see A132049/A132050 for the simplified fractions. - M. F. Hasler, Apr 03 2013

a(n+1) = sum of row n in triangle A008280. - Reinhard Zumkeller, Nov 05 2013

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon (2011) give a far-reaching generalization of the bijection between Euler numbers and alternating permutations. - N. J. A. Sloane, Jul 09 2015

D. André, Sur les permutations alternées, Journal de Mathématiques Pures et Appliquées, 7 (1881), 167-184.

Arnold, V. I., Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics, Duke Math. J. 63 (1991), 537-555.

M. D. Atkinson: Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.

M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, Feb 1985), Congressus Numerantium 47, 77-88.

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.

J. M. Borwein and S. T. Chapman, I prefer pi ..., Amer. Math. Monthly, 122 (2015), 195-216.

Brightwell, Graham; Cohen, Gerard; Fachini, Emanuela; Fairthorne, Marianne; Korner, Janos; Simonyi, Gabor; and Toth, Agnes Permutation capacities of families of oriented infinite paths. SIAM J. Discrete Math. 24 (2010), no. 2, 441-456.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 258-260, section #11.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262.

C. Davis, Problem 4755, Amer. Math. Monthly, 64 (1957) 596; solution by W. J. Blundon, 65 (1958), 533-534. [Denoted by P_n in solution.]

H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 66.

N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.

Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, Nov 20 2013; http://www-irma.u-strasbg.fr/~foata/paper/pub123Seidel.pdf

D. Foata and M.-P. Schutzenberger, Nombres d'Euler et permutations alternantes, in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.

Heinz-Richard Halder, Über Verfeinerungen von Partitionen, Periodica Mathematica Hungarica Vol. 12 (3), (1981), pp. 217-220.

O. Heimo and A. Karttunen, Series help-mates in 8, 9 and 10 moves (Problems 2901, 2974-2976), Suomen Tehtavaniekat (Proceedings of the Finnish Chess Problem Society) vol. 60, no. 2/2006, pp. 75, 77.

F. Heneghan and T. K. Petersen, Power series for up-down min-max permutations, http://math.depaul.edu/tpeter21/MaxMinUpDownCMJ2v.pdf, 2013.

L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 238.

G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.

F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 7 (1984), 191-199.

E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 110.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 184.

Remmel, Jeffrey B. Generating functions for alternating descents and alternating major index. Ann. Comb. 16 (2012), no. 3, 625--650. MR2960023. - From N. J. A. Sloane, Jan 15 2013

L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, vol. 7 (1877), 157-187.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Staib, Trigonometric power series, Math. Mag., 49 (1976), 147-148.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997 and Vol. 2, 1999; see Problem 5.7.

N. J. A. Sloane, The first 200 Euler numbers: Table of n, a(n) for n = 0..199

Joerg Arndt, Matters Computational (The Fxtbook), pp. 281-282

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.

P. Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.

F. Bergeron, M. Bousquet-Mélou and S. Dulucq, Standard paths in the composition poset

David Callan, A note on downup permutations and increasing 0-1-2 trees

Chandler Davis, Problem 4755: A Permutation Problem,  Amer. Math. Monthly, 64 (1957) 596; solution by W. J. Blundon, 65 (1958), 533-534. [Denoted by P_n in solution.] [Annotated scanned copy]

Filippo Disanto and Thomas Wiehe, Some combinatorial problems on binary rooted trees occurring in population genetics, arXiv preprint arXiv:1112.1295, 2011

Filippo Disanto, Andre' permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, arXiv preprint arXiv:1202.1139, 2012

R. Donaghey, Alternating permutations and binary increasing trees, J. Combinatorial Theory Ser. A 18 (1975), 141--148.MR0360299 (50 #12749)

O. Dovgoshey, E. Petrov, H.-M. Teichert, On spaces extremal for the Gomory-Hu inequality, arXiv preprint arXiv:1412.1979, 2014

D. Dumont & G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Preprint, Annotated scanned copy.

Richard Ehrenborg and N. Bradley Fox, The Descent Set Polynomial Revisited, arXiv:1408.6851, 2014. See Table 4.

N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf)

N. D. Elkies, New Directions in Enumerative Chess Problems, The Electronic Journal of Combinatorics, vol. 11(2), 2004.

P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function

S. N. Gladkovskii, On the continued fraction expansion for functions 1/sin(x) + cot(x) and sec(x) + tan(x),  arXiv:1208.2243v1 [math.HO]

W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-Mar 11 2013

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

B. R. Jones, On tree hook length formulae, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.

M. Josiat-Verges, Enumeration of snakes and cycle-alternating permutations

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

Dmitry Kruchinin, Integer properties of a composition of exponential generating functions, arXiv:1211.2100

Kruchinin, Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565

F. Luca and P. Stanica, On some conjectures on the monotonicity of some arithmetical sequences, J. Combin. Number Theory 4 (2012) 1-10.

J. M. Luck, On the frequencies of patterns of rises and falls, arXiv preprint arXiv:1309.7764, 2013

P. Luschny, Approximation, inclusion and asymptotics of the Euler numbers.

P. Luschny, An old operation on sequences: the Seidel transform

C. L. Mallows, Letter to N. J. A. Sloane, no date

J. L. Martin and J. D. Wagner, Updown numbers and the initial monomials of the slope variety, Electronic J. Combin. 16, no. 1 (2009), Research Paper R82. [From Jeremy L. Martin, Mar 26 2010]

A. Mendes, A note on alternating permutations, Amer. Math. Monthly, 114 (2007), 437-440.

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

D. J. Newman, W. Weissblum,  and others, Problem 67-5: "Up-Down" Permutations, Amer. Math. Monthly, Vol. ? (Year?), page 121, and Vol. ? (Year?), pages 225-226. [Annotated scanned copy]

E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411, 2013

A. Randrianarivony and J. Zeng, Sur une extension des nombres d'Euler et les records des permutations alternantes, J. Combin. Theory Ser. A 68 (1994), 68-99.

A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Y. Sano, The principal numbers of K. Saito for the types A_l, D_l and E_l, Discr. Math., 307 (2007), 2636-2642.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

R. P. Stanley, Queue problems revisited, Suomen Tehtavaniekat (Proceedings of the Finnish Chess Problem Society), vol. 59, no. 4 (2005), 193-203.

R. P. Stanley, Permutations

Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., 308 (2007), 71-112.

Ross Tang, An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series [From Ross Tang (ph.tchaa(AT)gmail.com), Jul 28 2010. Web page no longer accessible, pdf of archive.org version uploaded by Ralf Stephan, Dec 28 2013]

S. T. Thompson, Problem E754: Skew Ordered Sequences, Amer. Math. Monthly, 54 (1947), 416-417. [Annotated scanned copy]

A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938, 2011

Eric Weisstein's World of Mathematics, Euler Zigzag Number

Eric Weisstein's World of Mathematics, Alternating Permutation

Eric Weisstein's World of Mathematics, Entringer Number

Index entries for "core" sequences

Index entries for sequences related to boustrophedon transform

E.g.f.: (1+sin(x))/cos(x) = tan(x) + sec(x).

E.g.f. for a(n+1) is 1/(cos(x/2)-sin(x/2))^2 = 1/(1-sin(x)) = d/dx(sec(x)+tan(x)).

E.g.f. A(x)=-log(1-sin(x)), for a(n+1). - Vladimir Kruchinin, Aug 09 2010

O.g.f.: A(x) = 1+x/(1-x-x^2/(1-2*x-3*x^2/(1-3*x-6*x^2/(1-4*x-10*x^2/(1-... -n*x-(n*(n+1)/2)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna, Jan 17 2006

O.g.f. A(x) = y satisfies 2y' = 1 + y^2. - Michael Somos, Feb 03 2004

a(n) = P_n(0) + Q_n(0) (see A155100 and A104035), defining Q_{-1} = 0. Cf. A156142.

2*a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k).

Asymptotics: a(n) ~ 2^(n+2)*n!/Pi^(n+1).

a(n) = (n-1)*a(n-1) - sum{i=2, n-2, (i-1)*E(n-1, i)}, where E are the Entringer numbers A008280. - Jon Perry, Jun 09 2003

a(2*k) = (-1)^k euler(2k) and a(2k-1) = (-1)^(k-1)2^(2k)(2^(2k)-1) bernoulli(2k)/(2k). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005

|a(n+1)-2*a(n)| = A000708(n). - Philippe Deléham, Jan 13 2007

a(n) = 2^n|E(n,1/2)+E(n,1)| where E(n,x) are the Euler polynomials. - Peter Luschny, Jan 25 2009

a(n) = 2^{n+2}*n!*S(n+1)/(Pi)^{n+1}, where S(n)=Sum(1/(4k+1)^n, k=-inf..inf) (see the Elkies reference). - Emeric Deutsch, Aug 17 2009

a(n) = i^{n+1} sum_{k=1..n+1} sum_{j=0..k} binomial(k,j)(-1)^j (k-2j)^{n+1} (2i)^{-k} k^{-1}. - Ross Tang (ph.tchaa(AT)gmail.com), Jul 28 2010

a(n) = sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*stirling2(n,j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n) else 0),k,1,n), n>0. - Vladimir Kruchinin, Aug 19 2010

If n==1(mod 4) is prime, then a(n)==1(mod n); if n==3(mod 4) is prime, then a(n)==-1(mod n). - Vladimir Shevelev, Aug 31 2010

For m>=0, a(2^m)==1(mod 2^m); If p is prime, then a(2*p)==1(mod 2*p). - Vladimir Shevelev, Sep 03 2010

From Peter Bala, Jan 26 2011: (Start)

a(n) = A(n,I)/(1+I)^(n-1), where I = sqrt(-1) and {A(n,x)}n>=1 = [1,1+x,1+4*x+x^2,1+11*x+11*x^2+x^3,...] denotes the sequence of Eulerian polynomials.

Equivalently, a(n) = I^(n+1)*sum {k=1..n} (-1)^k*k!*Stirling2(n,k) * ((1+I)/2)^(k-1) = I^(n+1)*sum {k = 1..n} (-1)^k*((1+I)/2)^(k-1)* sum{j  = 0..k} (-1)^(k-j)*binomial(k,j)*j^n.

This explicit formula for a(n) can be used to obtain congruence results. For example, for odd prime p, a(p) = (-1)^((p-1)/2) (mod p), as noted by Vladimir Shevelev above.

For the corresponding type B results see A001586. For the corresponding results for plane increasing 0-1-2 trees see A080635.

For generalized Eulerian, Stirling and Bernoulli numbers associated with the zigzag numbers see A145876, A147315 and A185424, respectively. For a recursive triangle to calculate a(n) see A185414.

(End)

a(n) = I^(n+1)*2*Li_{-n}(-I) for n > 0. Li_{s}(z) is the polylogarithm. - Peter Luschny, Jul 29 2011

a(n) = 2*sum(m=0..(n-2)/2, 4^m*(sum(i=m..(n-1)/2, (i-(n-1)/2)^(n-1)*binomial(n-2*m-1,i-m)*(-1)^(n-i-1)))), n>1, a(0)=1, a(1)=1. - Vladimir Kruchinin, Aug 09 2011

E.g.f.: tan(x)+sec(x)=1+x/U(0); U(k)= 4k+1-x/(2-x/(4k+3+x/(2+x/U(k+1))));(continued fraction). - Sergei N. Gladkovskii, Nov 14 2011

a(n) = D^(n-1)(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1-x^2)*d/dx. Cf. A006154. a(n) equals the alternating sum of the nonzero elements of row n-1 of A196776. This leads to a combinatorial interpretation for a(n); for example, a(4*n+2) counts the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 1 (mod 4), minus the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 3 (mod 4). Cf A002017. - Peter Bala, Dec 06 2011

E.g.f. for a(n+1) is E(x)=1/(1-sin(x))=1 + x/(1 - x + x^2/G(0)); G(k)= (2*k+2)*(2*k+3)-x^2+(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 06 2012

E.g.f. for a(n+1) is E(x)=1/(1-sin(x))=1/(1 - x/(1 + x^2/G(0)) ; G(k)= 8*k+6-x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction Euler's 2 kind, 2-step). - Sergei N. Gladkovskii, Jan 06 2012

E.g.f. for a(n+1) is E(x)= 1/(1 - sin(x))=1/(1 - x*G(0)) ; G(k)= 1 - x^2/(2*(2*k+1)*(4*k+3) - 2*x^2*(2*k+1)*(4*k+3)/(x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction 3 kind, 3-step). - Sergei N. Gladkovskii, Jan 06 2012

E.g.f. for a(n+1) is E(x)= 1/(1 - sin(x))=1/(1 - x*G(0)) where G(k)= 1 - x^2/( (2*k+1)*(2*k+3) - (2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2) /G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 01 2012

E.g.f.: tan(x)+sec(x)=1+2*x/(U(0)-x) where U(k)= 4k+2 - x^2/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 27 2012

E.g.f.: tan(x)+sec(x) = 1+2*x/(2*U(0)-x) where U(k) = 4*k+1 - x^2/(16*k+12 - x^2/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 11 2012

E.g.f.: tan(x)+ sec(x) = 4/(2-x*G(0))-1 where G(k)=1 - x^2/(x^2 - 4*(2*k+1)*(2*k+3)/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012

G.f.: 1 + x/Q(0),m=+4,u=x/2, where Q(k) = 1 - 2*u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/( 1 - 2*u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013

E.g.f.: tan(x/2+Pi/4). - Vaclav Kotesovec, Nov 08 2013

G.f.: conjecture: 1 + T(0)*x/(1-x), where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013

E.g.f.: 1+ 4*x/(T(0) - 2*x), where T(k) = 4*(2*k+1) - 4*x^2/T(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 01 2013

E.g.f.: T(0)-1, where T(k) = 2 + x/(4*k+1 - x/(2 - x/( 4*k+3 + x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013

Asymptotic expansion: 4*(2*n/(Pi*e))^(n+1/2)*exp(1/2+1/(12*n)-1/(360*n^3) +1/(1260*n^5)-...). (See the Luschny link.) - Peter Luschny, Jul 14 2015

From Peter Bala, Sep 10 2015: (Start)

The e.g.f. A(x) = tan(x) + sec(x) satisfies A''(x) = A(x)*A'(x), hence the recurrence a(0) = 1, a(1) = 1, else a(n) = Sum_{i = 0..n-2} binomial(n-2,i)*a(i)*a(n-1-i).

Note, the same recurrence, but with the initial conditions a(0) = 0 and a(1) = 1, produces the sequence [0,1,0,1,0,4,0,34,0,496,...], an aerated version of A002105. (End)

G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 61*x^6 + 272*x^7 + 1385*x^8 + ...

Sequence starts 1,1,2,5,16,... since possibilities are {}, {A}, {AB}, {ACB, BCA}, {ACBD, ADBC, BCAD, BDAC, CDAB}, {ACBED, ADBEC, ADCEB, AEBDC, AECDB, BCAED, BDAEC, BDCEA, BEADC, BECDA, CDAEB, CDBEA, CEADB, CEBDA, DEACB, DEBCA}, etc. - Henry Bottomley, Jan 17 2001

A000111 := n-> n!*coeff(series(sec(x)+tan(x), x, n+1), x, n);

s := series(sec(x)+tan(x), x, 100): A000111 := n-> n!*coeff(s, x, n);

A000111:=n->piecewise(n mod 2=1, (-1)^((n-1)/2)*2^(n+1)*(2^(n+1)-1)*bernoulli(n+1)/(n+1), (-1)^(n/2)*euler(n)):seq(A000111(n), n=0..30); A000111:=proc(n) local k: k:=floor((n+1)/2): if n mod 2=1 then RETURN((-1)^(k-1)*2^(2*k)*(2^(2*k)-1)*bernoulli(2*k)/(2*k)) else RETURN((-1)^k*euler(2*k)) fi: end:seq(A000111(n), n=0..30); (C. Ronaldo)

T := n -> 2^n*abs(euler(n, 1/2)+euler(n, 1)): # Peter Luschny, Jan 25 2009

S := proc(n, k) option remember; if k=0 then RETURN(`if`(n=0, 1, 0)) fi; S(n, k-1)+S(n-1, n-k) end:

A000364 := n -> S(2*n, 2*n);

A000182 := n -> S(2*n+1, 2*n+1);

A000111 := n -> S(n, n); # Peter Luschny, Jul 29 2009

a := proc (n) options operator, arrow: 2^(n+2)*factorial(n)*(sum(1/(4*k+1)^(n+1), k = -infinity .. infinity))/Pi^(n+1) end proc: 1, seq(a(n), n = 1 .. 22); # Emeric Deutsch, Aug 17 2009

n=22; CoefficientList[Series[(1+Sin[x])/Cos[x], {x, 0, n}], x] * Table[k!, {k, 0, n}] (* Jean-François Alcover, May 18 2011, after Michael Somos *)

a[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1)*(2^(n+1)-1)*BernoulliB[n+1])/(n+1)]]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 09 2012, after C. Ronaldo *)

ee = Table[ 2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, 26}]; Table[ Differences[ee, n] // First // Abs, {n, 0, 26}] (* Jean-François Alcover, Mar 21 2013, after Paul Curtz *)

a[ n_] := If[ n < 0, 0, (2 I)^n If[ EvenQ[n], EulerE[n, 1/2], EulerE[n, 0] I]]; (* Michael Somos, Aug 15 2015 *)

a[ n_] := If[ n < 1, Boole[n == 0], With[{m = n - 1}, m! SeriesCoefficient[ 1 / (1 - Sin[x]), {x, 0, m}]]]; (* Michael Somos, Aug 15 2015 *)

From Michael Somos, Feb 03 2004: (Start)

(PARI) {a(n) = if( n<1, n==0, n--; n! * polcoeff( 1 / (1 - sin(x + x * O(x^n))), n))};

(PARI) {a(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])};

(PARI) {a(n) = local(an); if( n<1, n>=0, an = vector(n+1, m, 1); for( m=2, n, an[m+1] = sum( k=0, m-1, binomial(m-1, k) * an[k+1] * an[m-k]) / 2); an[n+1])};

(End)

(PARI) z='z+O('z^66); egf = (1+sin(z))/cos(z); Vec(serlaplace(egf)) \\ Joerg Arndt, Apr 30 2011

(PARI) A000111(n)={my(k); sum(m=0, n\2, (-1)^m*sum(j=0, k=n+1-2*m, binomial(k, j)*(-1)^j*(k-2*j)^(n+1))/k>>k)}  \\ M. F. Hasler, May 19 2012

(PARI) A000111(n)=if(n, 2*abs(polylog(-n, I)), 1)  \\ M. F. Hasler, May 20 2012

(Maxima) a(n):=sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*stirling2(n, j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n) else 0), k, 1, n); /* Vladimir Kruchinin, Aug 19 2010 */

(Maxima)

a(n):=if n<2 then 1 else 2*sum(4^m*(sum((i-(n-1)/2)^(n-1)*binomial(n-2*m-1, i-m)*(-1)^(n-i-1), i, m, (n-1)/2)), m, 0, (n-2)/2); /* Vladimir Kruchinin, Aug 09 2011 */

(Sage) # Algorithm of L. Seidel (1877)

def A000111_list(n) :

    R = []; A = {-1:0, 0:1}; k = 0; e = 1

    for i in (0..n) :

        Am = 0; A[k + e] = 0; e = -e

        for j in (0..i) : Am += A[k]; A[k] = Am; k += e

        R.append(A[-i//2] if i%2 == 0 else A[i//2])

    return R

A000111_list(22) # Peter Luschny, Mar 31 2012

(Haskell)

a000111 0 = 1

a000111 n = sum $ a008280_row (n - 1)

-- Reinhard Zumkeller, Nov 01 2013

(Python)

# requires python 3.2 or higher

from itertools import accumulate

A000111_list, blist = [1, 1], [1]

for n in range(10**2):

....blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))

....A000111_list.append(sum(blist)) # Chai Wah Wu, Jan 29 2015

Cf. A000364 (secant numbers), A000182 (tangent numbers). See also A008280, A008281, A008282, A010094, A059720 for related triangles.

A diagonal of A008970.

Cf. A109449 for an extension to an exponential Riordan array.

Column k=1 of A229892, A258829.

Column k=2 of A250261. Cf. A002105.

Sequence in context: A178123 A138265 A163747 * A007976 A058259 A033543

Adjacent sequences:  A000108 A000109 A000110 * A000112 A000113 A000114

nonn,core,eigen,nice,easy,changed

N. J. A. Sloane

Edited by M. F. Hasler, Apr 04 2013

Title corrected by Geoffrey Critzer, May 18 2013

approved