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

Author

Seiichi Manyama, Mar 19 2025

Keywords

• nonn
• tabf

Examples

			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;
  ...
		

Links

• 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.

Crossrefs

Row sums give A048617.

Cf. A125300, A165889, A173018.

Programs

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), ", ")));

Formula

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