cp's OEIS Frontend

A373578 Expansion of e.g.f. exp(x * (1 + x^2)^2).

%I A373578 #20 Jun 11 2024 08:09:45
%S A373578 1,1,1,13,49,241,2401,13021,128353,1346689,10615681,140431501,
%T A373578 1544877841,17576665393,264566466529,3226728670621,48376006929601,
%U A373578 766753039205761,11052669865900033,197019825098096269,3271213100827557361,56597110823949654001
%N A373578 Expansion of e.g.f. exp(x * (1 + x^2)^2).
%H A373578 Vaclav Kotesovec, <a href="/A373578/b373578.txt">Table of n, a(n) for n = 0..500</a>
%F A373578 a(n) = n! * Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k)/(n-2*k)!.
%F A373578 a(n) == 1 (mod 12).
%F A373578 a(n) = a(n-1) + 6*(n-1)*(n-2)*a(n-3) + 5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5).
%F A373578 a(n) ~ 5^(n/5 - 1/2) * exp(7*5^(-11/5)*n^(1/5) + 2*5^(-3/5)*n^(3/5) - 4*n/5) * n^(4*n/5). - _Vaclav Kotesovec_, Jun 11 2024
%t A373578 nmax = 20; CoefficientList[Series[E^(x*(1 + x^2)^2), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 11 2024 *)
%o A373578 (PARI) a(n) = n!*sum(k=0, 2*n\5, binomial(2*n-4*k, k)/(n-2*k)!);
%Y A373578 Cf. A118395, A190863, A373577.
%Y A373578 Cf. A361278.
%K A373578 nonn
%O A373578 0,4
%A A373578 _Seiichi Manyama_, Jun 10 2024