

A160563


Table of the number of (n,k)Riordan complexes, read by rows.


1



1, 1, 1, 9, 10, 1, 225, 259, 35, 1, 11025, 12916, 1974, 84, 1, 893025, 1057221, 172810, 8778, 165, 1, 108056025, 128816766, 21967231, 1234948, 28743, 286, 1, 18261468225, 21878089479, 3841278805, 230673443, 6092515, 77077, 455, 1, 4108830350625, 4940831601000
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OFFSET

0,4


COMMENTS

From Table 4, righthand side, of Gelineau and Zeng. The JacobiStirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the LegendreStirling numbers.
Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the JacobiStirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.
Essentially a rowreversal of A008956.  R. J. Mathar, May 20 2009


LINKS

Table of n, a(n) for n=0..37.
Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the JacobiStirling Numbers, arXiv:0905.2899 [math.CO], May 2009.
W. Zhang, Some identities involving the Euler and the central factorial numbers, The Fibonacci Quarterly, Vol. 36, Number 2, May 1998.


FORMULA

a(n,k) = v(n,k) where v(n,k) = v(n1,k1)  (2n1)^2*v(n1,k); eq (4.2).
Let F(x) = 1/cos(x). Then (2*n)!*(1/cos(x))^(2*n+1) = Sum_{k=0..n} T(n,k)*F^(2*k)(x), where F^(r) denotes the rth derivative of F(x) (Zhang 1998). An example is given below.  Peter Bala, Feb 06 2012


EXAMPLE

For row 3: F(x) := 1/cos(x). Then 225*F(x) + 259*(d/dx)^2(F(x)) + 35*(d/dx)^4(F(x)) + (d/dx)^6(F(x)) = 720*(1/cos(x))^7, where F^(r) denotes the rth derivative of F(x).


MAPLE

t := proc(n, k) option remember ; expand(x*mul(x+n/2i, i=1..n1)) ; coeftayl(%, x=0, k) ; end:
v := proc(n, k) option remember ; 4^(nk)*t(2*n+1, 2*k+1) ; end:
A160563 := proc(n, k) abs(v(n, k)) ; end: for n from 0 to 10 do for k from 0 to n do printf("%d, ", A160563(n, k)) ; od: od: # R. J. Mathar, May 20 2009


MATHEMATICA

t[_, 0] = 1; t[n_, n_] := t[n, n] = ((2*n  1)!!)^2; t[n_, k_] := t[n, k] = (2*n  1)^2*t[n  1, k  1] + t[n  1, k];
T[n_, k_] := t[n, n  k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Nov 28 2017, after R. J. Mathar's comment *)


CROSSREFS

Cf. A001819, A008275, A008277, A160562, A091885.
Sequence in context: A324663 A109409 A262551 * A158286 A126839 A341818
Adjacent sequences: A160560 A160561 A160562 * A160564 A160565 A160566


KEYWORD

nonn,tabl


AUTHOR

Jonathan Vos Post, May 19 2009


EXTENSIONS

Extended by R. J. Mathar, May 20 2009


STATUS

approved



