OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
E.g.f. A(x) satisfies A(x) = (1 + 2*x*A(x)^5)^(1/2).
a(n) = 2^n * n! * binomial((5*n+1)/2,n)/(5*n+1).
a(n) = Sum_{k=0..n} (4*n+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ 5^(5*n/2) * n^(n-1) / (exp(n) * 3^(3*n/2 + 1)). - Vaclav Kotesovec, Jul 04 2025
MATHEMATICA
a[n_]:=If[n==0, 1, Product[(5*n+1-2*k), {k, 0, n-1}]/(5*n+1)]; Table[a[n], {n, 0, 18}] (* Vincenzo Librandi, Mar 27 2026 *)
PROG
(PARI) a(n) = 2^n*n!*binomial((5*n+1)/2, n)/(5*n+1);
(Magma) [n eq 0 select 1 else (2^n*&*[((5*n+1)/2-k): k in [0..n-1] ])/(5*n+1): n in [0..23]]; // Vincenzo Librandi, Mar 27 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 29 2025
STATUS
approved
