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
Keywords
Programs
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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!));
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PARI
a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(-3*i+4)*v[i]+3*(i-1)*v[i-1]+1); v;
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