cp's OEIS Frontend

A361278 Expansion of e.g.f. exp(x * (1+x)^2).

%I A361278 #23 Nov 11 2023 08:21:54
%S A361278 1,1,5,19,97,581,3661,26335,202049,1659817,14621941,135567851,
%T A361278 1326672865,13624218349,146056961597,1633376573431,18980051829121,
%U A361278 228677164878545,2852155973178469,36740599423566787,488127224550517601,6678832987859315221
%N A361278 Expansion of e.g.f. exp(x * (1+x)^2).
%H A361278 Winston de Greef, <a href="/A361278/b361278.txt">Table of n, a(n) for n = 0..566</a>
%F A361278 a(n) = n! * Sum_{k=0..n} binomial(2*k,n-k)/k!.
%F A361278 a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(2,k-1) * a(n-k)/(n-k)!.
%F A361278 D-finite with recurrence a(n) -a(n-1) +4*(-n+1)*a(n-2) -3*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Mar 12 2023
%F A361278 a(n) ~ 3^(n/3 - 1/2) * n^(2*n/3) / exp(2*n/3 - 2*3^(-2/3)*n^(2/3) - 3^(-7/3)*n^(1/3) + 4/81) * (1 + 953*3^(1/3)/(4374*n^(1/3)) - 2051059*3^(2/3)/(191318760*n^(2/3))). - _Vaclav Kotesovec_, Nov 11 2023
%p A361278 A361278 := proc(n)
%p A361278     n!*add(binomial(2*k,n-k)/k!,k=0..n) ;
%p A361278 end proc:
%p A361278 seq(A361278(n),n=0..60) ; #_R. J. Mathar_, Mar 12 2023
%o A361278 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^2)))
%o A361278 (PARI) a(n) = n!*sum(k=0, n, binomial(2*k, n-k)/k!);
%o A361278 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(2, j-1)*v[i-j+1]/(i-j)!)); v;
%Y A361278 Column k=2 of A361277.
%Y A361278 Cf. A082579.
%K A361278 nonn
%O A361278 0,3
%A A361278 _Seiichi Manyama_, Mar 06 2023