A361279 - OEIS

%I #25 Nov 11 2023 08:48:42

%S 1,1,7,37,241,1981,17551,171697,1860097,21609721,268697431,3566446621,

%T 50060084977,740156116597,11496472967071,186824483634601,

%U 3167058238988161,55882288483846897,1023891003620741287,19440027237549627541,381822392009503555441

%N Expansion of e.g.f. exp(x * (1+x)^3).

%H Winston de Greef, <a href="/A361279/b361279.txt">Table of n, a(n) for n = 0..508</a>

%F a(n) = n! * Sum_{k=0..n} binomial(3*k,n-k)/k!.

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(3,k-1) * a(n-k)/(n-k)!.

%F D-finite with recurrence a(n) -a(n-1) +6*(-n+1)*a(n-2) -9*(n-1)*(n-2)*a(n-3) -4*(n-1)*(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Mar 13 2023

%F a(n) ~ 2^(n/2 - 1) * n^(3*n/4) / exp(3*n/4 - 3*n^(3/4)/2^(3/2) - 15*n^(1/2)/64 + n^(1/4)/2^(19/2) + 27/1024) * (1 + 724053*sqrt(2)/(2621440*n^(1/4))). - _Vaclav Kotesovec_, Nov 11 2023

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^3)))

%o (PARI) a(n) = n!*sum(k=0, n, binomial(3*k, n-k)/k!);

%o (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(3, j-1)*v[i-j+1]/(i-j)!)); v;

%Y Column k=3 of A361277.

%Y Cf. A091695.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 06 2023