A379877 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^2) + x*A(x)^3.

1, 0, 1, 2, 33, 244, 4345, 61830, 1332961, 28087208, 739562481, 20380504330, 644853623425, 21767589641628, 810480865644073, 32246095869576974, 1385625666085792065, 63366863108725330000, 3090966367543869021409, 159607809547688836085778, 8718178798812199357657441

Offset: 0

Author

Seiichi Manyama, Jan 05 2025

Keywords

• nonn

Links

• Kummer Functions, Digital Library of Mathematical Functions, Jan. 2025.

Crossrefs

Cf. A379871, A379876, A379878.

Programs

Mathematica

Table[-n! * Sum[(-2*n - 1)^(n-k-1) * Binomial[2*n + k, k]/(n-k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2025 *)

PARI

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

Formula

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

a(n) = U(-n, -3*n, -1 - 2*n)/(1 + 2*n), where U is the Kummer U function. - David Trimas, Jan 09 2025

a(n) ~ 2^(3*n) * n^(n-1) / (sqrt(3) * exp(2*n + 1/2)). - Vaclav Kotesovec, Jan 15 2025