A353546 Expansion of e.g.f. -log(1-2*x) * exp(x)/2.
0, 1, 4, 17, 96, 729, 7060, 83033, 1146656, 18164625, 324488068, 6450956929, 141233271872, 3376008830505, 87480173354964, 2442396780039817, 73089894980585408, 2333809837398044321, 79198287879591647364, 2846319497398561356913
Offset: 0
Keywords
Programs
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PARI
my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-2*x)*exp(x)/2))) -
PARI
a(n) = n!*sum(k=0, n-1, 2^(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]=(2*i-1)*v[i]-2*(i-1)*v[i-1]+1); v;
Formula
a(n) = n! * Sum_{k=0..n-1} 2^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (2 * n - 1) * a(n-1) - 2 * (n-1) * a(n-2) + 1.
a(n) ~ (n-1)! * exp(1/2) * 2^(n-1). - Vaclav Kotesovec, Jun 08 2022