A353549 Expansion of e.g.f. log(1+3*x) * exp(x)/3.

0, 1, -1, 12, -104, 1289, -19605, 356488, -7541464, 182009385, -4935863537, 148600324124, -4918093868688, 177482897072545, -6936155749635541, 291836667412104072, -13152940374866178512, 632196357654491385521, -32280617841842744380161

Offset

0,4

Links

Table of n, a(n) for n=0..18.

Formula

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

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

a(n) ~ -(-1)^n * (n-1)! * 3^(n-1) / exp(1/3). - Vaclav Kotesovec, Jun 08 2022

Prog

(PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(log(1+3*x)*exp(x)/3)))
(PARI) a(n) = n!*sum(k=0, n-1, (-3)^(n-1-k)/((n-k)*k!));
(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;

Crossrefs

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

Cf. A346398.

Sequence in context: A173835 A334168 A220111 * A094530 A157024 A051815

Adjacent sequences: A353546 A353547 A353548 * A353550 A353551 A353552

Keyword

sign

Author

Seiichi Manyama, May 27 2022

Status

approved