A361283 - OEIS

%I #17 Nov 11 2023 10:16:22

%S 1,1,9,85,961,13041,207001,3746149,75832065,1693615681,41302616041,

%T 1090835399061,30988423000129,941461990360945,30439632977968761,

%U 1042973073239321701,37731609890300935681,1436586994020158747649

%N Expansion of e.g.f. exp(x/(1-x)^4).

%H Seiichi Manyama, <a href="/A361283/b361283.txt">Table of n, a(n) for n = 0..414</a>

%F a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(-4*k,n-k)/k! = n! * Sum_{k=0..n} binomial(n+3*k-1,n-k)/k!.

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

%F D-finite with recurrence a(n) +(-5*n+4)*a(n-1) +(n-1)*(10*n-23)*a(n-2) -10*(n-1)*(n-2)*(n-3)*a(n-3) +5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Mar 12 2023

%F a(n) ~ 2^(1/5) * n^(n - 1/10) * exp(-27/1280 - 13*2^(3/5)*n^(1/5)/800 + 13*2^(1/5)*n^(2/5)/240 + 2^(-6/5)*n^(3/5) + 5*2^(-8/5)*n^(4/5) - n) / sqrt(5) * (1 + 116303*2^(12/5)/(3200000*n^(1/5))). - _Vaclav Kotesovec_, Nov 11 2023

%p A361283 := proc(n)

%p     n!*add(binomial(n+3*k-1,n-k)/k!,k=0..n) ;

%p end proc:

%p seq(A361283(n),n=0..40) ; # _R. J. Mathar_, Mar 12 2023

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^4)))

%o (PARI) a(n) = n!*sum(k=0, n, (-1)^(n-k)*binomial(-4*k, n-k)/k!);

%o (PARI) a(n) = n!*sum(k=0, n, binomial(n+3*k-1, 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, (-1)^(j-1)*j*binomial(-4, j-1)*v[i-j+1]/(i-j)!)); v;

%Y Column k=4 of A293012.

%Y Cf. A361280.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 06 2023