A353549 - OEIS

%I #15 Jun 08 2022 09:56:10

%S 0,1,-1,12,-104,1289,-19605,356488,-7541464,182009385,-4935863537,

%T 148600324124,-4918093868688,177482897072545,-6936155749635541,

%U 291836667412104072,-13152940374866178512,632196357654491385521,-32280617841842744380161

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

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

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

%F a(n) ~ -(-1)^n * (n-1)! * 3^(n-1) / exp(1/3). - _Vaclav Kotesovec_, Jun 08 2022

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(log(1+3*x)*exp(x)/3)))

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

%Y Cf. A002104, A353546, A353547, A353548, A354419.

%Y Cf. A346398.

%K sign

%O 0,4

%A _Seiichi Manyama_, May 27 2022