OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..395
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * (n-3*k+1)^(n-3*k-1) * binomial(2*(n-3*k),k)/(n-3*k)!.
E.g.f.: exp( -LambertW(-x * (1-x^3)^2) ).
MATHEMATICA
a[n_]:=n!*Sum[(-1)^k*(n-3*k+1)^(n-3*k-1)*Binomial[2*(n-3*k), k]/(n-3*k)!, {k, 0, Floor[n/3]}]; Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 25 2025 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (-1)^k*(n-3*k+1)^(n-3*k-1)*binomial(2*(n-3*k), k)/(n-3*k)!);
(Magma) a := func< n | Factorial(n) * &+[ (-1)^k * (n-3*k + 1)^(n-3*k - 1) * Binomial(2*(n-3*k), k) / Factorial(n-3*k) : k in [0..Floor(n/3)]] >;
[a(n) : n in [0..25]]; // Vincenzo Librandi, Oct 25 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 22 2025
STATUS
approved
