OFFSET
0,3
COMMENTS
See A326323 for the more general formulas.
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
KEYWORD
nonn
AUTHOR
Peter Luschny, Jun 27 2019
EXTENSIONS
Corrected after notice from Jean-François Alcover by Peter Luschny, Jul 13 2019
STATUS
approved