A389869 E.g.f. A(x) satisfies A(x) = exp( x^3 * A(x) / (1-x) ) / (1-x).

1, 1, 2, 12, 96, 840, 9000, 118440, 1794240, 30542400, 583027200, 12360902400, 287500752000, 7278608937600, 199453032979200, 5882801987462400, 185803266391142400, 6257404402221772800, 223868182248610099200, 8479703733055614259200, 339031061570199398400000

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)^2) )/(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, A389870.

Sequence in context: A306258 A382411 A052691 * A371040 A292419 A322543

Adjacent sequences: A389866 A389867 A389868 * A389870 A389871 A389872

Keyword

nonn

Author

Seiichi Manyama, Oct 18 2025

Status

approved