A171150 - OEIS

A171150

Triangle related to T(x,2x).

1, 1, 1, 2, 3, 1, 3, 9, 7, 1, 6, 20, 28, 15, 1, 10, 50, 85, 75, 31, 1, 20, 105, 255, 294, 186, 63, 1, 35, 245, 651, 1029, 903, 441, 127, 1, 70, 504, 1736, 3108, 3612, 2568, 1016, 255, 1, 126, 1134, 4116, 9324, 12636, 11556, 6921, 2295, 511, 1, 252, 2310, 10290, 25080, 42120, 46035, 34605, 17930, 5110, 1023, 1

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OFFSET

0,4

COMMENTS

Let the triangle T_(x,y)=T defined by T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1.

This triangle gives the coefficients of Sum_{k=0..n} T(n,k) where y=2x.

T_(0,0) = A053121, T_(1,2) = A039599, T_(2,4) = A124575.

First column of T_(x,2x) is given by A126222.

LINKS

Table of n, a(n) for n=0..65.

M. Barnabei, F. Bonetti, and M. Silimbani, The Eulerian numbers on restricted centrosymmetric permutations, PU. M. A. Vol. 21 (2010), No. 2, pp. 99-118 (see Table p. 118, with additional zeros); see also, arXiv:0910.2376 [math.CO], 2009.

FORMULA

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001405(n), A000984(n), A133158(n) for x = -1, 0, 1, 2 respectively.

EXAMPLE

Triangle begins:

1, 1;

2, 3, 1;

3, 9, 7, 1;

6, 20, 28, 15, 1;

10, 50, 85, 75, 31, 1;

...

CROSSREFS

Cf. A000012, A000225, A058877, A126222.