A389870 E.g.f. A(x) satisfies A(x) = exp( x^3 * A(x) / (1-x)^3 ) / (1-x).

1, 1, 2, 12, 144, 1920, 27000, 425880, 7842240, 167348160, 4021315200, 106544592000, 3084311260800, 97118951529600, 3310090700716800, 121414524725894400, 4767198569099366400, 199481309606663884800, 8863477218614501683200, 416838113104377213849600

Offset

0,3

Links

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

Formula

E.g.f.: exp( -LambertW(-x^3/(1-x)^4) )/(1-x).

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

Mathematica

Table[n!*Sum[(k+1)^(k-1)*Binomial[n+k, n-3*k]/k!, {k, 0, Floor[n/3]}], {n, 0, 25}] (* Vincenzo Librandi, Oct 28 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*binomial(n+k, n-3*k)/k!);
(Magma) [Factorial(n) * &+[(k+1)^(k-1)* Binomial(n+k, n-3*k) / Factorial(k) : k in [0..Floor(n/3)]] : n in [0..25] ]; // Vincenzo Librandi, Oct 28 2025

Crossrefs

Cf. A371039, A378095, A389869.

Cf. A389866, A389868.

Sequence in context: A208866 A334174 A372993 * A067601 A365284 A052740

Adjacent sequences: A389867 A389868 A389869 * A389871 A389872 A389873

Keyword

nonn

Author

Seiichi Manyama, Oct 18 2025

Status

approved