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

0,3

Comments

See A326323 for the more general formulas.

Links

Table of n, a(n) for n=0..19.

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

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}]

Crossrefs

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

Sequence in context: A261499 A001829 A006386 * A215084 A232058 A331704

Adjacent sequences: A326321 A326322 A326323 * A326325 A326326 A326327

Keyword

nonn

Author

Peter Luschny, Jun 27 2019

Extensions

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

Status

approved