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A039755 Triangle of B-analogues of Stirling numbers of the second kind. 21
1, 1, 1, 1, 4, 1, 1, 13, 9, 1, 1, 40, 58, 16, 1, 1, 121, 330, 170, 25, 1, 1, 364, 1771, 1520, 395, 36, 1, 1, 1093, 9219, 12411, 5075, 791, 49, 1, 1, 3280, 47188, 96096, 58086, 13776, 1428, 64, 1, 1, 9841, 239220, 719860, 618870, 209622, 32340, 2388, 81, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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 Bn 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)

LINKS

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

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]

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]

FORMULA

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)

EXAMPLE

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

MAPLE

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) ; [From R. J. Mathar, Oct 30 2009]

MATHEMATICA

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-Fran├žois Alcover, Jun 09 2011, after Peter Bala *)

PROG

(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))

CROSSREFS

Cf. A154537, A214406.

Sequence in context: A140070 A158815 A101275 * A047874 A080248 A139382

Adjacent sequences:  A039752 A039753 A039754 * A039756 A039757 A039758

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified April 19 03:59 EDT 2014. Contains 240738 sequences.