A380674 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x * (1 - x)^2) ).

1, 3, 25, 370, 8097, 237096, 8733601, 388380000, 20253654945, 1212334652800, 81937521020841, 6172429566120192, 512850795552978625, 46594245206418954240, 4595466275857015549425, 488993161791784338804736, 55839856392986843905585089, 6811561624203525171739852800

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))^2)/(1 - x*A(x))^2.

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

a(n) ~ 3^(3*n+3) * n^(n-1) / (sqrt(23) * 2^(2*n + 3/2) * exp((23*n-4)/27)). - Vaclav Kotesovec, Feb 01 2026

Mathematica

Table[n! * Sum[(n+1)^(k-1) * Binomial[3*n-3*k+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-3*k+1, n-k)/k!);

Crossrefs

Cf. A377832, A380665, A380666, A380675.

Sequence in context: A394478 A377829 A295765 * A012481 A132617 A241703

Adjacent sequences: A380671 A380672 A380673 * A380675 A380676 A380677

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2025

Status

approved