A380665 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x/(1 - x)) ).

1, 3, 31, 586, 16401, 612336, 28678231, 1618268688, 106946168769, 8105456425600, 693228400344591, 66055574392722432, 6940237183385667409, 797165049089377683456, 99381018789002592800775, 13365207839280075801020416, 1928719845703457066672384769, 297293268794967068206087176192

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Index entries for reversions of series

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x)))/(1 - x*A(x))^2.

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(3*n+1,n-k)/k!.

a(n) ~ 2^(n + 1/2) * 3^(3*n + 3/2) * n^(n-1) / (13^(1/4) * (11 - sqrt(13))^(n + 1/2) * exp(((5 - sqrt(13))*n + 3 - sqrt(13))/2)). - Vaclav Kotesovec, Feb 01 2026

Mathematica

Table[n! * Sum[(n+1)^(k-1) * Binomial[3*n+1, n-k] / k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 01 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(3*n+1, n-k)/k!);

Crossrefs

Cf. A052873, A380663.

Cf. A377832.

Sequence in context: A273378 A266487 A229258 * A382015 A319896 A370260

Adjacent sequences: A380662 A380663 A380664 * A380666 A380667 A380668

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2025

Status

approved