A389696 E.g.f. A(x) satisfies A(x) = 1 + x*exp(x^2)*A(x)^3.

1, 1, 6, 78, 1464, 37140, 1190880, 46222680, 2107728000, 110462682960, 6543131616000, 432313593798240, 31523415971604480, 2514671753061948480, 217847820118560952320, 20367541513745880854400, 2044119892615204884480000, 219194951926263548749113600, 25011190475780053581324288000

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * A001764(n-2*k)/k!.

a(n) ~ sqrt(3*(1 + LambertW(32/729))) * 2^((n-3)/2) * n^(n-1) / (exp(n) * LambertW(32/729)^(n/2)). - Vaclav Kotesovec, Nov 13 2025

Mathematica

Join[{1}, Table[n!*Sum[(n - 2*k)^k * Binomial[3*n - 6*k, n - 2*k]/(2*n - 4*k + 1)/k!, {k, 0, Floor[n/2]}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 13 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(3*(n-2*k), n-2*k)/((2*(n-2*k)+1)*k!));
(Magma) [Factorial(n)*&+[(n-2*k)^k*Binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1)/Factorial(k): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 29 2025

Crossrefs

Cf. A358064, A387714.

Cf. A364983, A390611.

Cf. A001764.

Sequence in context: A131926 A132866 A279444 * A094419 A229044 A300874

Adjacent sequences: A389693 A389694 A389695 * A389697 A389698 A389699

Keyword

nonn

Author

Seiichi Manyama, Nov 12 2025

Status

approved