A353548 - OEIS

%I #15 Jul 17 2022 23:27:28

%S 0,1,6,47,540,8429,166210,3952955,109981816,3502905369,125648153278,

%T 5011458069639,219987094389524,10538817637744005,547118005892177018,

%U 30595552548140425747,1833501625083035349488,117219490267316310468913

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

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

%F a(0) = 0, a(1) = 1, a(n) = (4 * n - 3) * a(n-1) - 4 * (n-1) * a(n-2) + 1.

%F a(n) ~ (n-1)! * exp(1/4) * 4^(n-1). - _Vaclav Kotesovec_, Jun 08 2022

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-4*x)*exp(x)/4)))

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

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=0; v[2]=1; for(i=2, n, v[i+1]=(4*i-3)*v[i]-4*(i-1)*v[i-1]+1); v;

%Y Cf. A002104, A353546, A353547, A353549.

%Y Cf. A346396.

%Y Essentially partial sums of A056545.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 27 2022