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A354290
Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(3 - 2*exp(x)).
3
1, 2, 14, 142, 1878, 30494, 585398, 12946910, 323717622, 9020101470, 276940926646, 9283709731806, 337237965060982, 13191050077634654, 552593521885522486, 24677110613547498718, 1169994350288769049334, 58684818937875321715038
OFFSET
0,2
FORMULA
a(0) = 1; a(n) = Sum_{k=1..n} A004123(k+1) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 2^k * A000262(k) * Stirling2(n,k).
a(n) ~ exp(1/(6*log(3/2)) - 5/6 + 2*sqrt(n)/sqrt(3*log(3/2)) - n) * (n^(n - 1/4) / (sqrt(2) * 3^(1/4) * log(3/2)^(n + 1/4))). - Vaclav Kotesovec, May 23 2022
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1)/(3-2*exp(x)))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 2^k*k!*stirling(j, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 23 2022
STATUS
approved