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

0,4

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,  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.

Row sums give A000984.

Sequence in context: A152440 A134319 A135091 * A111589 A259760 A010027

Adjacent sequences: A171147 A171148 A171149 * A171151 A171152 A171153

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 04 2009

Extensions

More terms from Alois P. Heinz, Jan 31 2023

Status

approved