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%I #18 Jun 08 2022 09:58:53
%S 0,1,-2,23,-276,4509,-91190,2205587,-62104168,1995807993,-72089029802,
%T 2891304481999,-127498010037244,6131189086886421,-319320539953144158,
%U 17905976286288568267,-1075611833288214177232,68909527979479961534705
%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 + 5) * a(n-1) + 4 * (n-1) * a(n-2) + 1.
%F a(n) ~ -(-1)^n * (n-1)! * 4^(n-1) / exp(1/4). - _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+5)*v[i]+4*(i-1)*v[i-1]+1); v;
%Y Cf. A002104, A353546, A353547, A353548, A353549.
%K sign
%O 0,3
%A _Seiichi Manyama_, May 27 2022