|
%I #13 May 23 2022 10:05:27
%S 1,2,14,142,1878,30494,585398,12946910,323717622,9020101470,
%T 276940926646,9283709731806,337237965060982,13191050077634654,
%U 552593521885522486,24677110613547498718,1169994350288769049334,58684818937875321715038
%N Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(3 - 2*exp(x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} A004123(k+1) * binomial(n-1,k-1) * a(n-k).
%F a(n) = Sum_{k=0..n} 2^k * A000262(k) * Stirling2(n,k).
%F 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
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1)/(3-2*exp(x)))))
%o (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;
%Y Cf. A075729, A354291.
%Y Cf. A004123, A354286, A354288.
%K nonn
%O 0,2
%A _Seiichi Manyama_, May 23 2022
|
