A390014 E.g.f. A(x) satisfies A(x) = exp( x * (1-x^3) * A(x) ).

1, 1, 3, 16, 101, 936, 11047, 157144, 2666169, 52078672, 1150141931, 28347264144, 770603470837, 22903367650216, 738840293877519, 25708877220859816, 959844376755705713, 38274296147683826976, 1623490328244862405459, 72992587093309269078688, 3467473188638256590907501

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..390

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * (n-3*k+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.

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

Mathematica

a[n_]:=n!*Sum[(-1)^k*(n-3*k+1)^(n-3*k-1)*Binomial[n-3*k, k]/(n-3*k)!, {k, 0, Floor[n/4]}]; Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 24 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\4, (-1)^k*(n-3*k+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);
(Magma) a := func< n | Factorial(n) * &+[ (-1)^k * (n-3*k + 1)^(n-3*k - 1) * Binomial(n-3*k, k) / Factorial(n-3*k) : k in [0..Floor(n/4)] ] >;
[a(n) : n in [0..25]]; // Vincenzo Librandi, Oct 24 2025

Crossrefs

Cf. A389104, A390013.

Sequence in context: A009007 A394123 A000949 * A390215 A091637 A278429

Adjacent sequences: A390011 A390012 A390013 * A390015 A390016 A390017

Keyword

nonn

Author

Seiichi Manyama, Oct 22 2025

Status

approved