A382086 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
1, 1, 5, 52, 845, 18816, 533617, 18404800, 748039833, 35016198400, 1855389108221, 109781344134144, 7174844881882405, 513331696318615552, 39905830821183755625, 3349445733955326754816, 301886246619209909215793, 29080090017105458412257280, 2981488457660004727761477493
Offset: 0
Keywords
Programs
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PARI
a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(n+k-1, k)/(n-k-1)!));
Formula
E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(n+k-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x) * exp(-x) ) ).
a(n) ~ phi^(3*n - 3/2) * n^(n-1) / (5^(1/4) * exp((n - 1/phi)/phi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 15 2025