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%I #14 Feb 04 2022 10:36:37
%S 1,1,6,46,456,5656,84336,1467376,29175936,652606336,16219458816,
%T 443419545856,13224580002816,427278468668416,14867050125981696,
%U 554245056343668736,22039796215883268096,931198483176870608896,41658202699736550014976,1967160945260218035798016
%N a(n) = A_{5}(n) where A_{m}(x) are the Eulerian polynomials as defined in A326323.
%C See A326323 for the more general formulas.
%F a(n) ~ n!/5 * (4/log(5))^(n+1). - _Vaclav Kotesovec_, Aug 09 2021
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k). - _Ilya Gutkovskiy_, Feb 04 2022
%p seq(add(combinat:-eulerian1(n,k)*5^k, k=0..n), n=0..20);
%p # Alternative:
%p egf := 4/(5 - exp(4*x)): ser := series(egf, x, 21):
%p seq(k!*coeff(ser, x, k), k=0..20);
%t a[1] := 1; a[n_] := 4^(n + 1)/5 HurwitzLerchPhi[1/5, -n, 0];
%t Table[a[n], {n, 0, 20}]
%t (* Alternative: *)
%t s[n_] := Sum[StirlingS2[n, j] 4^(n - j) j!, {j, 0, n}];
%t Table[s[n], {n, 0, 20}]
%Y Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326323.
%K nonn
%O 0,3
%A _Peter Luschny_, Jun 27 2019
%E Corrected after notice from _Jean-François Alcover_ by _Peter Luschny_, Jul 13 2019
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