A326324 a(n) = A_{5}(n) where A_{m}(x) are the Eulerian polynomials as defined in A326323.

1, 1, 6, 46, 456, 5656, 84336, 1467376, 29175936, 652606336, 16219458816, 443419545856, 13224580002816, 427278468668416, 14867050125981696, 554245056343668736, 22039796215883268096, 931198483176870608896, 41658202699736550014976, 1967160945260218035798016

Offset: 0

Author

Peter Luschny, Jun 27 2019

Keywords

• nonn

Comments

See A326323 for the more general formulas.

Crossrefs

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326323.

Programs

Maple

seq(add(combinat:-eulerian1(n,k)*5^k, k=0..n), n=0..20);
# Alternative:
egf := 4/(5 - exp(4*x)): ser := series(egf, x, 21):
seq(k!*coeff(ser, x, k), k=0..20);

Mathematica

a[1] := 1; a[n_] := 4^(n + 1)/5 HurwitzLerchPhi[1/5, -n, 0];
Table[a[n], {n, 0, 20}]
(* Alternative: *)
s[n_] := Sum[StirlingS2[n, j] 4^(n - j) j!, {j, 0, n}];
Table[s[n], {n, 0, 20}]

Formula

a(n) ~ n!/5 * (4/log(5))^(n+1). - Vaclav Kotesovec, Aug 09 2021

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k). - Ilya Gutkovskiy, Feb 04 2022

Extensions

Corrected after notice from Jean-François Alcover by Peter Luschny, Jul 13 2019