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

0,3

Links

Table of n, a(n) for n=0..18.

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

Prog

(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)!));

Crossrefs

Cf. A382087, A382088, A382089.

Cf. A000108, A377831, A382036.

Sequence in context: A367165 A381444 A377324 * A357346 A196531 A108459

Adjacent sequences: A382083 A382084 A382085 * A382087 A382088 A382089

Keyword

nonn

Author

Seiichi Manyama, Mar 15 2025

Status

approved