A377554 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^3 ).

1, 3, 30, 537, 14124, 493695, 21601458, 1137294039, 70064934600, 4947238170747, 394022075650590, 34951812094581723, 3417754921150904172, 365287875167708973831, 42368411854713294141834, 5300422308901745571018735, 711465905597330333014408848, 101995745742232833085109746803

Offset

0,2

Links

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

Index entries for reversions of series

Formula

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^3.

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364986.

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

a(n) ~ s*(1 + r*s) * n^(n-1) / (sqrt(2/3 + r*s^(2/3)*(2 + r*s)*(1 + s^(1/3) + r*s)) * exp(n) * r^n), where r = 0.1088452788470987813073112378046285091604229890225... and s = 2.490387060898719218398726253445677946092717961367... are roots of the system of equations (1 + exp(r*s)*r*s)^3 = s, 3*exp(r*s)*r*(1 + r*s)*s^(2/3) = 1. - Vaclav Kotesovec, Feb 05 2026

Mathematica

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

Crossrefs

Cf. A161633, A377553.

Cf. A364986.

Sequence in context: A185827 A366009 A370910 * A375871 A080527 A370057

Adjacent sequences: A377551 A377552 A377553 * A377555 A377556 A377557

Keyword

nonn

Author

Seiichi Manyama, Nov 01 2024

Status

approved