A396434 Expansion of (F_2(x)/x)^(1/3), where F_k(x) is the k-th iteration of x*G4(x)^3 with G4(x) = 1 + x*G4(x)^4.

1, 2, 12, 94, 836, 8025, 81136, 852012, 9210604, 101884076, 1148162064, 13138949028, 152296165996, 1784568913542, 21106289095896, 251636459582520, 3021119305369280, 36494117062138956, 443229504130411376, 5409100104838981832, 66296660011662002352, 815719349332409935946

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..895

Wikipedia, Fuss-Catalan number

Formula

G.f.: ((1/x) * Series_Reversion( H_2(x) ))^(1/3), where H_k(x) is the k-th iteration of x*(1 - x)^3.

a(n) = Sum_{k=0..n} (3*k+1) * binomial(4*k+1,k) * binomial(4*n-k+1,n-k)/((4*k+1) * (4*n-k+1)).

Prog

(PARI)
lista(nn, k=2, p=4, s=3, r=1) = {
  my(T=matrix(nn+1, nn+1, row, col, my(xr=row-1, xc=col-1); if(xc<xr, 0, (s*xr+r)*binomial(p*xc-(p-s)*xr+r, xc-xr)/(p*xc-(p-s)*xr+r))));
  my(TK=T^k);
  TK[1, ];
};

Crossrefs

Column k=2 of A396448.

Sequence in context: A316143 A372202 A359692 * A321057 A366334 A247075

Adjacent sequences: A396431 A396432 A396433 * A396435 A396436 A396437

Keyword

nonn

Author

Seiichi Manyama, May 25 2026

Status

approved