A377831 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x) ).
1, 2, 13, 154, 2701, 63216, 1856569, 65711024, 2724349401, 129552751360, 6952877604421, 415770771875328, 27416031835737637, 1976460653044957184, 154658036515292528625, 13055394531339601033216, 1182611605875201470044081, 114426900236922150187892736
Offset: 0
Keywords
Programs
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PARI
a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n-k, n-k)/k!);
Formula
E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n-k,n-k)/k!.
a(n) ~ phi^(3*n + 3/2) * n^(n-1) / (5^(1/4) * exp((n+1)/phi - 1)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 09 2024