A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

1, 2, 13, 217, 5937, 223641, 10725433, 625007993, 42883208609, 3386452550689, 302545287708201, 30170153462509545, 3322052185576104049, 400328811249634307249, 52406094009429908677049, 7405663486143907784247481, 1123601498350780798756198209, 182173718779147621454796872769

Offset: 0

Author

Seiichi Manyama, Mar 11 2025

Keywords

• nonn

Crossrefs

Cf. A381984, A381986.

Cf. A001764, A002293, A346646, A364987.

Programs

PARI

a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(4*k+1, k)/((4*k+1)*(n-k)!));

Formula

Let F(x) be the e.g.f. of A364987. F(x) = B(x*A(x)) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002293(k)/(n-k)!.