0,5

Let M = an infinite lower triangular bidiagonal matrix with (1,3,5,7,...) in the main diagonal and (1,1,1,...) in the subdiagonal. n-th row = M^n * [1,0,0,0,...]. - Gary W. Adamson, Apr 13 2009

From Peter Bala, Aug 08 2011: (Start)

A type B_n set partition is a partition P of the set {1, 2, . . . , n, -1, -2, . . . , -n} such that for any block B of P, -B is also a block of P, and there is at most one block, called a zero-block, satisfying B = -B. We call (B, -B) a block pair of P if B is not a zero-block. Then T(n,k) is the number of type B_n set partitions with k block pairs. See [Wang].

For example, T(2,1) = 4 since the B_2 set partitions with 1 block pair are {1,2}{-1,-2}, {1,-2}{-1,2}, {1,-1}{2}{-2} and {2,-2}{1}{-1} (the last two partitions contain a zero block).

(End)

Exponential Riordan array [exp(x),1/2(exp(2*x) - 1)]. Triangle of connection constants for expressing the monomial polynomials x^n as a linear combination of the basis polynomials (x-1)*(x-3)*...*(x-(2*k-1)) of A039757. An example is given below. Inverse array is A039757. Equals matrix product A008277 * A122848. - Peter Bala, Jun 23 2014

Table of n, a(n) for n=0..55.

P. Bala, Generalized Dobinski formulas

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.

Sandrine Dasse-Hartaut and Pawel Hitczenko, Greek letters in random staircase tableaux arXiv:1202.3092v1 [math.CO]

L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207v5 [math.CO]

Shi-Mei Ma, T. Mansour, D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv preprint arXiv:1404.0731, 2014

R. Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

D. G. L. Wang, The Limiting Distribution of the Number of Block Pairs in Type B Set Partitions, arXiv:1108.1264v1 [math.CO]

E.g.f./G.f.: exp(x + y/2 * (exp(2*x) - 1)).

T(n,k) = T(n-1,k-1)+(2*k+1)*T(n-1,k) with T(0,0)=T(1,0)=T(1,1)=0. sum_{k=0..n} T(n,k) = A007405(n). - R. J. Mathar, Oct 30 2009

T(n,k) = 1/(2^k*k!) * sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2*j+1)^n.

T(n,k) = 1/(2^k*k!) * A145901(n,k). - Peter Bala

The row polynomials R(n,x) satisfy the Dobinski-type identity:

R(n,x) = exp(-x/2)* sum {k >= 0} (2*k+1)^n*(x/2)^k/k!, as well as the recurrence equation R(n+1,x) = (1+x)*R(n,x)+2*x*R'(n,x). The polynomial R(n,x) has all real zeros (apply [Liu et al., Theorem 1.1] with f(x) = R(n,x) and g(x) = R'(n,x)). The polynomials R(n,2*x) are the row polynomials of A154537. - Peter Bala, Oct 28 2011

Let f(x) = exp(1/2*exp(2*x)+x). Then the row polynomials R(n,x) are given by R(n,exp(2*x)) = 1/f(x)*(d/dx)^n(f(x)). Similar formulas hold for A008277, A105794, A111577, A143494 and A154537. - Peter Bala, Mar 01 2012

From Peter Bala, Jul 20 2012: (Start)

The o.g.f. for the n-th diagonal (with interpolated zeros) is the rational function D^n(x), where D is the operator x/(1-x^2)*d/dx. For example, D^3(x) = x*(1+8*x^2+3*x^4)/(1-x^2)^5 = x + 13*x^3 + 58*x^5 + 170*x^7 + ... . See A214406 for further details.

An alternative formula for the o.g.f. of the n-th diagonal is exp(-x/2)*(sum {k >= 0} (2*k+1)^(k+n-1)*(x/2*exp(-x))^k/k!).

(End)

Triangle T(n,k) begins:

1

1   1

1   4   1

1  13   9   1

1  40  58  16  1

1 121 330 170 25 1

The sequence of row polynomials of A214406 begins [1, 1+x, 1+8*x+3*x^2, ...]. The o.g.f.'s for the diagonals of this triangle thus begin

1/(1-x) = 1 + x + x^2 + x^3 + ...

(1+x)/(1-x)^3 = 1 + 4*x + 9*x^2 + 16*x^3 + ...

(1+8*x+3*x^2)/(1-x)^5 = 1 + 13*x + 58*x^2 + 170*x^3 + ... . - Peter Bala, Jul 20 2012

Connection constants: x^3 = 1 + 13*(x-1) + 9*(x-1)*(x-3) + (x-1)*(x-3)*(x-5). Hence row 3 = [1,13,9,1]. - Peter Bala, Jun 23 2014

A039755 := proc(n, k) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n-1, k-1)+(2*k+1)*procname(n-1, k) ; fi; end: seq(seq(A039755(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Oct 30 2009

t[n_, k_] = Sum[(-1)^(k-j)*(2j+1)^n*Binomial[k, j], {j, 0, k}]/(2^k*k!); Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[1 ;; 56]]

(* Jean-François Alcover, Jun 09 2011, after Peter Bala *)

(PARI) T(n, k)=if(k<0|k>n, 0, n!*polcoeff(polcoeff(exp(x+y/2*(exp(2*x+x*O(x^n))-1)), n), k))

Cf. A154537, A214406, A039756, A039757, A122848.

Sequence in context: A140070 A158815 A101275 * A247502 A047874 A080248

Adjacent sequences:  A039752 A039753 A039754 * A039756 A039757 A039758

nonn,tabl

Ruedi Suter (suter(AT)math.ethz.ch)

approved