A160563 - OEIS

A160563

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

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, right-hand side, of Gelineau and Zeng.

Essentially a row-reversal 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 Jacobi-Stirling 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(n-1,k-1) - (2n-1)^2*v(n-1,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 r-th derivative of F(x) (Zhang 1998). An example is given below. - Peter Bala, Feb 06 2012

Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=1..n step 2, n=1..len step 2] the 'odd subtriangle' of U. This triangle is the odd subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit with signed terms. See A182867 for the even subtriangle. - Peter Luschny, Mar 03 2024

EXAMPLE

Triangle starts:

[0] 1;

[1] 1, 1;

[2] 9, 10, 1;

[3] 225, 259, 35, 1;

[4] 11025, 12916, 1974, 84, 1;

[5] 893025, 1057221, 172810, 8778, 165, 1;

[6] 108056025, 128816766, 21967231, 1234948, 28743, 286, 1;

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 r-th derivative of F(x).

MAPLE

t := proc(n, k) option remember ; expand(x*mul(x+n/2-i, i=1..n-1)) ; coeftayl(%, x=0, k) ; end:

v := proc(n, k) option remember ; 4^(n-k)*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

# Using a bivariate generating function (albeit generating signed terms):

gf := (t + sqrt(1 + t^2))^x: ser := series(gf, t, 20):

ct := n -> coeff(ser, t, n): T := (n, k) -> n!*coeff(ct(n), x, k):

OddPart := (T, len) -> local n, k;

seq(print(seq(T(n, k), k = 1..n, 2)), n = 1..2*len, 2):

OddPart(T, 6); # Peter Luschny, Mar 03 2024

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 (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar's comment *)

CROSSREFS

Cf. A001819, A008275, A008277, A160562, A091885, A182867.