A392855 E.g.f. A(x) satisfies A(x) = 1/(1 - x*(exp(x*A(x)^2)-1)^2).

1, 0, 0, 6, 24, 70, 3780, 60914, 612528, 17541486, 526589340, 11012612842, 313737059592, 11974631476262, 395860489923444, 13812862050432930, 599628896186261856, 26293293080734483678, 1152424587118462645068, 56907888879265236839258, 3000511798650573548923320

Offset

0,4

Links

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (2*k)! * binomial(2*n-k+1,k)/(2*n-k+1) * Stirling2(n-k,2*k)/(n-k)!.

Mathematica

Table[n!* Sum[(2*k)!*Binomial[2*n-k+1, k]/(2*n-k+1)*StirlingS2[n-k, 2*k]/(n-k)!, {k, 0, Floor[n/3]}], {n, 0, 20}] (* Vincenzo Librandi, Feb 06 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (2*k)!*binomial(2*n-k+1, k)/(2*n-k+1)*stirling(n-k, 2*k, 2)/(n-k)!);
(Magma) [Factorial(n) * &+[Factorial(2*k)* Binomial(2*n-k+1, k)/(2*n-k+1) * StirlingSecond(n - k, 2*k) / Factorial(n - k): k in [0..Floor(n/3)]]: n in [0..25] ]; // Vincenzo Librandi, Feb 06 2026

Crossrefs

Cf. A392820, A392825.

Cf. A392768.

Sequence in context: A392826 A392825 A392766 * A379825 A337021 A262445

Adjacent sequences: A392852 A392853 A392854 * A392856 A392857 A392858

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved