A361280 Expansion of e.g.f. exp(x * (1+x)^4).

1, 1, 9, 61, 481, 4881, 55321, 682669, 9343041, 139078081, 2216425321, 37736834301, 683184324769, 13064452686481, 262867726142841, 5549111222344621, 122499654278797441, 2819926900630750209, 67539541277010100681, 1679557316488693881661

Offset

0,3

Links

Winston de Greef, Table of n, a(n) for n = 0..476

Formula

a(n) = n! * Sum_{k=0..n} binomial(4*k,n-k)/k!.

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(4,k-1) * a(n-k)/(n-k)!.

D-finite with recurrence a(n) -a(n-1) +8*(-n+1)*a(n-2) -18*(n-1)*(n-2)*a(n-3) -16*(n-1)*(n-2)*(n-3)*a(n-4) -5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 12 2023

a(n) ~ 5^(n/5 - 1/2) * n^(4*n/5) * exp(-256/15625 - 249*5^(4/5)*n^(1/5)/78125 + 236*5^(3/5)*n^(2/5)/9375 + 22*5^(2/5)*n^(3/5)/125 + 4*5^(-4/5)*n^(4/5) - 4*n/5) * (1 + 15409886*5^(1/5)/(48828125*n^(1/5))). - Vaclav Kotesovec, Nov 11 2023

Maple

A361280 := proc(n)
    n!*add(binomial(4*k, n-k)/k!, k=0..n) ;
end proc:
seq(A361280(n), n=0..60) ; # R. J. Mathar, Mar 12 2023

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^4)))
(PARI) a(n) = n!*sum(k=0, n, binomial(4*k, n-k)/k!);
(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(4, j-1)*v[i-j+1]/(i-j)!)); v;

Crossrefs

Column k=4 of A361277.

Cf. A361283.

Sequence in context: A200674 A162769 A126504 * A025014 A246567 A322086

Adjacent sequences: A361277 A361278 A361279 * A361281 A361282 A361283

Keyword

nonn

Author

Seiichi Manyama, Mar 06 2023

Status

approved