

A039755


Triangle of Banalogs of Stirling numbers of the second kind.


22



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

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. nth 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 zeroblock, satisfying B = B. We call (B, B) a block pair of P if B is not a zeroblock. 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 (x1)*(x3)*...*(x(2*k1)) of A039757. An example is given below. Inverse array is A039757. Equals matrix product A008277 * A122848.  Peter Bala, Jun 23 2014


LINKS

Table of n, a(n) for n=0..55.
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
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 DasseHartaut and Pawel Hitczenko, Greek letters in random staircase tableaux arXiv:1202.3092v1 [math.CO], 2012.
I. Dolgachev and V. Lunts, A character formula for the representation of the Weyl group in the cohomology of the associated toric variety Journal of Algebra, 168, 741772, (1994)
L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207v5 [math.CO], 20052006.
ShiMei Ma, T. Mansour, D. Callan, Some combinatorial arrays related to the LotkaVolterra system, arXiv preprint arXiv:1404.0731 [math.CO], 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], 2011.


FORMULA

E.g.f./g.f.: exp(x + y/2 * (exp(2*x)  1)).
T(n,k) = T(n1,k1) + (2*k+1)*T(n1,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)^(kj)*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 Dobinskitype 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 nth diagonal (with interpolated zeros) is the rational function D^n(x), where D is the operator x/(1x^2)*d/dx. For example, D^3(x) = x*(1+8*x^2+3*x^4)/(1x^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 nth diagonal is exp(x/2)*(Sum_{k >= 0} (2*k+1)^(k+n1)*(x/2*exp(x))^k/k!).
(End)
From Tom Copeland, Dec 31 2015: (Start)
T(n,m) = Sum_{i=0..nm} 2^(nmi)*binomial(n,i)*St2(ni,m), where St2(n,k) are the Stirling numbers of the second kind, A048993 (also A008277). See p. 755 of Dolgachev and Lunts.
The relation of this entry's e.g.f. above to that of the Bell polynomials, Bell_n(y), of A048993 establishes this formula from a binomial transform of the normalized Bell polynomials, NB_n(y) = 2^n Bell_n(y/2); that is, e^x exp[(y/2)(e^(2x)1)] = e^x exp[x*2*Bell.(y/2)] = exp[x(1+NB.(y))] = exp(x*P.(y)), so the row polynomials of this entry are given by P_n(y) = [1+NB.(y)]^n = sum(k=0,..n) C(n,k) NB_k(y) = sum(k=0..n) 2^k C(n,k) Bell_k(y/2).
The umbral compositional inverses of the Bell polynomials are the falling factorials Fct_n(y) = y! / (yn)!; i.e., Bell_n(Fct.(y)) = y^n = Fct_n(Bell.(y)). Since P_n(y) = [1+2Bell.(y/2)]^n, the umbral inverses are determined by [1 + 2 Bell.[ 2 Fct.[(y1)/2] / 2 ] ]^n = [1 + 2 Bell.[ Fct.[(y1)/2] ] ]^n = [1+y1]^n = y^n. Therefore, the umbral inverse sequence of this entry's row polynomials is the sequence IP_n( y) = 2^n Fct_n[(y1)/2] = (y1)(y3) .. (y2n+1) with IP_0(y) = 1 and, from the binomial theorem, with e.g.f. exp[x IP.(y)]= exp[ x 2Fct.[(y1)/2] ] = (1+2x)^[(y1)/2] = exp[ [(y1)/2] log(1+2x) ].


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/(1x) = 1 + x + x^2 + x^3 + ...
(1+x)/(1x)^3 = 1 + 4*x + 9*x^2 + 16*x^3 + ...
(1+8*x+3*x^2)/(1x)^5 = 1 + 13*x + 58*x^2 + 170*x^3 + ... .  Peter Bala, Jul 20 2012
Connection constants: x^3 = 1 + 13*(x1) + 9*(x1)*(x3) + (x1)*(x3)*(x5). Hence row 3 = [1,13,9,1].  Peter Bala, Jun 23 2014


MAPLE

A039755 := proc(n, k) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n1, k1)+(2*k+1)*procname(n1, k) ; fi; end: seq(seq(A039755(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Oct 30 2009


MATHEMATICA

t[n_, k_] = Sum[(1)^(kj)*(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]]
(* JeanFrançois Alcover, Jun 09 2011, after Peter Bala *)


PROG

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


CROSSREFS

Cf. A154537, A214406, A039756, A039757, A122848.
Cf. A008277, A048993.
Sequence in context: A158815 A101275 A262494 * A247502 A047874 A080248
Adjacent sequences: A039752 A039753 A039754 * A039756 A039757 A039758


KEYWORD

nonn,tabl


AUTHOR

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


STATUS

approved



