0,3

Related to partially ordered sets. - Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003

The number of partial permutation matrices P in GL_n with P^2=0. Alternatively, the number of orbits of the Borel group of upper triangular matrices acting by conjugation on the set of matrices M in GL_n with M^2=0. - Brian Rothbach (rothbach(AT)math.berkeley.edu), Apr 16 2004

Number of ways to use the elements of {1,..,n} once each to form a collection of sequences, each having length 1 or 2. - Bob Proctor, Apr 18 2005

Hankel transform is A108400. - Paul Barry, Feb 11 2008

This is also the number of subsets of equivalent ways to arrange the elements of n pairs, when equivalence is defined under the joint operation of (optional) reversal of elements combined with permutation of the labels and the subset maps to itself. - Ross Drewe, Mar 16 2008

Equals inverse binomial transform of A000898 [From Gary W. Adamson, Oct 06 2008]

a(n) is also the moment of order n for the measure of density exp(-(x-1)^2/4)/(2*sqrt(Pi)) over the interval -infinity..infinity - Groux Roland, Mar 26 2011

The n-th term gives the number of fixed-point-free involutions in S_n^B, the group of permutations on the set {-n,...,-1,1,2,...,n}. - Matt Watson, Jul 26 2012

a(n) is also I^(-n)*H_{n}(I/2) with I the imaginary unit and H_{n} the Hermite polynomial of degree n. - Alyssa Byrnes and C. Vignat, Jan 31 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200

T. Amdeberhan, V. de Angelis, A. Dixit, V. H. Moll and C. Vignat, From sequences to polynomials and back, via operator orderings, J. Math. Phys. 54, 123502 (2013); Alternative copy

Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts); Alternative copy

Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394; Alternative copy.

Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926, 2011.

T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150, 2013

A. Khruzin, Enumeration of chord diagrams, arXiv:math.CO/0008209

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv math.CO.0606404.

Index entries for related partition-counting sequences

Index entries for sequences related to Hermite polynomials

seq( sum('n!/((n-2*k)!*k!)', 'k'=0..floor(n/2)), n=0..30 ); - Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001

E.g.f.: exp(x^2+x). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 11 2001

Binomial transform of A001813 (with interpolated zeros). - Paul Barry, May 09 2003

a(n)=sum{k=0..n, C(k,n-k)*n!/k!}; - Paul Barry, Mar 29 2007

a(n)=sum{k=0..floor(n/2), C(n,2k)(2k)!/k!}; - Paul Barry, Feb 11 2008

G.f.: 1/(1-x-2*x^2/(1-x-4*x^2/(1-x-6*x^2/(1-x-8*x^2/(1-... (continued fraction). [Paul Barry, Apr 10 2009]

E.g.f.: Q(0); Q(k)=1+(x^2+x)/(2*k+1-(x^2+x)*(2*k+1)/((x^2+x)+(2*k+2)/Q(k+1)))) ; (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A000085 and A115329. - Peter Bala, Dec 07 2011

a(n) ~ 2^(n/2-1/2)*exp(sqrt(n/2)-n/2-1/8)*n^(n/2). - Vaclav Kotesovec, Oct 08 2012

E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013

E.g.f.: -Q(0)/x where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013

G.f.: 1/Q(0), where Q(k)= 1 + x*2*k - x/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013

E.g.f.: E(0)-1-x-x^2, where E(k) = 2 + 2*x*(1+x) - 8*k^2 + x^2*(1+x)^2*(2*k+3)*(2*k-1)/E(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013

with(combstruct):seq(count(([S, {S=Set(Union(Z, Prod(Z, Z)))}, labeled], size=n)), n=0..30); # Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003

Range[0, 23]!*CoefficientList[ Series[ Exp[x*(1-x^2)/(1 - x)], {x, 0, 23 }], x] - (* Zerinvary Lajos, Mar 23 2007 *)

Table[I^(-n)*HermiteH[n, I/2], {n, 0, 23}] - (* Alyssa Byrnes and C. Vignat, Jan 31 2013 *)

(MATLAB) N = 18; A = zeros(N, 1); for n = 1:N; a = factorial(n); s = 0; k = 0; while k <= floor(n/2); b = factorial(n - 2*k); c = factorial(k); s = s + a/(b*c); k = k+1; end; A(n) = s; end; disp(A); - Ross Drewe, Mar 16 2008

(PARI) x='x+O('x^66); Vec(serlaplace(exp(x^2+x))) \\ Joerg Arndt, May 04 2013

Row sums of A067147.

Cf. A000680, A001147, A132101.

Cf. A000898.

Sequence in context: A148734 A124425 A118398 * A148735 A148736 A148737

Adjacent sequences:  A047971 A047972 A047973 * A047975 A047976 A047977

nonn

N. J. A. Sloane.

approved