A382232 Irregular triangle read by rows: T(n,k) = [x^k] (1+x) * A_n(x)^2, where A_n(x) is the n-th Eulerian polynomial.

1, 1, 1, 1, 1, 3, 3, 1, 1, 9, 26, 26, 9, 1, 1, 23, 165, 387, 387, 165, 23, 1, 1, 53, 860, 4292, 9194, 9194, 4292, 860, 53, 1, 1, 115, 3967, 38885, 160778, 314654, 314654, 160778, 38885, 3967, 115, 1, 1, 241, 17022, 307454, 2291375, 8041695, 14743812, 14743812, 8041695, 2291375, 307454, 17022, 241, 1

Offset

0,6

Links

Table of n, a(n) for n=0..57.

Ryuichi Sakamoto, The h*-polynomial of the cut polytope of K_{2,m} in the lattice spanned by its vertices, arXiv:1904.10667 [math.CO], 2019.

Ryuichi Sakamoto, The h*-polynomial of the cut polytope of K_{2,m} in the lattice spanned by its vertices, Journal of Integer Sequences, Vol. 23, 2020, #20.7.5.

OEIS Wiki, Eulerian polynomials.

Formula

T(n,k) = T(n,2*n-1-k) for n > 0.

Example

Irregular triangle begins:
  1,  1;
  1,  1;
  1,  3,   3,    1;
  1,  9,  26,   26,    9,    1;
  1, 23, 165,  387,  387,  165,   23,   1;
  1, 53, 860, 4292, 9194, 9194, 4292, 860, 53, 1;
  ...

Prog

(PARI) a(n) = sum(k=0, n, k!*stirling(n, k, 2)*(x-1)^(n-k));
T(n, k) = polcoef((1+x)*a(n)^2, k);
for(n=0, 7, for(k=0, 2*(n+0^n)-1, print1(T(n, k), ", ")));

Crossrefs

Row sums give A048617.

Cf. A125300, A165889, A173018.

Sequence in context: A340934 A271706 A172108 * A220666 A385432 A104378

Adjacent sequences: A382229 A382230 A382231 * A382233 A382234 A382235

Keyword

nonn,tabf

Author

Seiichi Manyama, Mar 19 2025

Status

approved