OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)^k * A001764(n-3*k)/k!.
a(n) ~ sqrt(1 + LambertW(64/6561)) * 3^(n/3 + 1/2) * n^(n-1) / (2^(3/2) * exp(n) * LambertW(64/6561)^(n/3)). - Vaclav Kotesovec, Nov 13 2025
MATHEMATICA
Join[{1}, Table[n!*Sum[(n - 3*k)^k * Binomial[3*n - 9*k, n - 3*k]/(2*n - 6*k + 1)/k!, {k, 0, Floor[n/3]}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 13 2025 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(3*(n-3*k), n-3*k)/((2*(n-3*k)+1)*k!));
(Magma) [Factorial(n)*&+[(n-3*k)^k*Binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1)/Factorial(k): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 29 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Nov 12 2025
STATUS
approved
