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

Offset: 0

Author

Philippe Deléham, Dec 04 2009

Keywords

• nonn
• tabl

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.

Examples

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   3,  9,  7,  1;
   6, 20, 28, 15,  1;
  10, 50, 85, 75, 31,  1;
  ...
		

Links

• 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.

Crossrefs

Cf. A000012, A000225, A058877, A126222.

Row sums give A000984.

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.

Extensions

More terms from Alois P. Heinz, Jan 31 2023