A364981 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^3).

1, 1, 4, 39, 580, 11685, 298566, 9248701, 336886936, 14112113049, 668422303210, 35325208755441, 2060811941835780, 131547166492534117, 9120279070776381886, 682489450793082237285, 54828316394224735284016, 4706545644403274325580593

Offset

0,3

Links

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*n-2*k+1,k)/( (3*n-2*k+1)*(n-k)! ).

a(n) ~ sqrt((1 + r*s^3)/(12*s + 9*r*s^4)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.1811100305436879929789759231994897963241226689807... and s = 1.522012903517407628213363540403002787906223513104... are real roots of the system of equations 1 + exp(r*s^3)*r*s = s, 3*r*s^3*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

Mathematica

Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[3*n-2*k+1, k] / ((3*n-2*k+1)*(n-k)!), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 18 2023 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*n-2*k+1, k)/((3*n-2*k+1)*(n-k)!));

Crossrefs

Cf. A006153, A161633, A364938, A364980.

Sequence in context: A323323 A300188 A177775 * A192935 A365010 A024055

Adjacent sequences: A364978 A364979 A364980 * A364982 A364983 A364984

Keyword

nonn

Author

Seiichi Manyama, Aug 15 2023

Status

approved