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 * Catalan(n-3*k)/k!.
E.g.f.: 2/(1 + sqrt(1 - 4*x*exp(x^3))).
a(n) ~ sqrt(2*(1 + LambertW(3/64))) * 3^(n/3) * n^(n-1) / (exp(n) * LambertW(3/64)^(n/3)). - Vaclav Kotesovec, Nov 13 2025
MATHEMATICA
Join[{1}, Table[Factorial[n]*Sum[(n-3*k)^k*Binomial[2*(n-3*k), n-3*k]/((n-3*k+1)*Factorial[k]), {k, 0, Floor[n/3]}], {n, 1, 20}]] (* Vincenzo Librandi, Dec 27 2025 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(2*(n-3*k), n-3*k)/((n-3*k+1)*k!));
(Magma) [Factorial(n)*&+[(n-3*k)^k*Binomial(2*(n-3*k), n-3*k)/((n-3*k+1)*Factorial(k)): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 27 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Nov 12 2025
STATUS
approved
