A377553 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^2 ).

1, 2, 14, 174, 3176, 77010, 2336892, 85316714, 3644408336, 178412603778, 9851421767060, 605826315779322, 41068369222584024, 3042849619010389058, 244657525386435161756, 21217387476442659806250, 1974219906922046702054432, 196191093901292764305110274, 20739322455031604846405387556

Offset

0,2

Links

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

Index entries for reversions of series

Formula

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364982.

a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(2*n+2,k)/(n-k)!.

a(n) ~ s*(1 + r*s) * n^(n-1) / (sqrt(r*sqrt(s)*(2 + r*s)*(1 + sqrt(s) + r*s) + 1/2) * exp(n) * r^n), where r = 0.1570317521587425394104145546878277976331805967621... and s = 2.44975698906826108253790826816096590467198031755... are roots of the system of equations (1 + exp(r*s)*r*s)^2 = s, 2*exp(r*s)*r*(1 + r*s)*s^(1/2) = 1. - Vaclav Kotesovec, Feb 05 2026

Mathematica

Join[{1}, Table[n! * Sum[k^(n-k)*Binomial[2*n+2, k]/(n-k)!/(n+1), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 02 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*n+2, k)/(n-k)!)/(n+1);

Crossrefs

Cf. A161633, A377554.

Cf. A364982.

Sequence in context: A167014 A381480 A370909 * A379576 A366736 A300282

Adjacent sequences: A377550 A377551 A377552 * A377554 A377555 A377556

Keyword

nonn

Author

Seiichi Manyama, Nov 01 2024

Status

approved