OFFSET
0,4
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * (exp(x) - 1)^2.
E.g.f.: 2/(1 + sqrt(1-4*x*(exp(x) - 1)^2)).
a(n) = n! * Sum_{k=0..floor(n/3)} (2*k)! * binomial(2*k+1,k)/(2*k+1) * Stirling2(n-k,2*k)/(n-k)!.
MATHEMATICA
Table[n!*Sum[(2*k)!*Binomial[2*k+1, k] /(2*k+1)*Abs[StirlingS2[n-k, 2*k]/(n-k)!], {k, 0, Floor[n/3]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 25 2026 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\3, (2*k)!*binomial(2*k+1, k)/(2*k+1)*stirling(n-k, 2*k, 2)/(n-k)!);
(Magma) [Factorial(n)* &+[Factorial(2*k)*Binomial(2*k+1, k)/(2*k+1)*StirlingSecond(n-k, 2*k)/Factorial(n-k) : k in [0..Floor(n/3)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 25 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 24 2026
STATUS
approved