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A039755
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Triangle of B-analogues of Stirling numbers of the second kind.
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17
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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
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OFFSET
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0,5
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COMMENTS
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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,...]. [From 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)
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REFERENCES
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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. - From N. J. A. Sloane, Dec 27 2012
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LINKS
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Table of n, a(n) for n=0..55.
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]
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FORMULA
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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). [From 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). [From 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 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)
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EXAMPLE
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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
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MAPLE
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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]
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MATHEMATICA
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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]]
(* From Jean-François Alcover, Jun 9 2011, after P. Bala *)
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PROG
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(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))
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CROSSREFS
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A154537. A214406.
Sequence in context: A140070 A158815 A101275 * A047874 A080248 A139382
Adjacent sequences: A039752 A039753 A039754 * A039756 A039757 A039758
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KEYWORD
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nonn,tabl
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AUTHOR
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Ruedi Suter (suter(AT)math.ethz.ch)
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STATUS
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approved
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