A396433 Expansion of (F_2(x)/x)^(1/2), where F_k(x) is the k-th iteration of x*G4(x)^2 with G4(x) = 1 + x*G4(x)^4.

1, 2, 11, 79, 645, 5688, 52850, 510147, 5070522, 51581908, 534750083, 5631311687, 60088612174, 648403965361, 7064553448007, 77615909028006, 858980155363199, 9567496484998948, 107170256222238701, 1206531521792903644, 13644565395364715036, 154931191054173346846

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..924

Wikipedia, Fuss-Catalan number

Formula

G.f.: ((1/x) * Series_Reversion( H_2(x) ))^(1/2), where H_k(x) is the k-th iteration of x*(1 - x*C(x))^2 with C(x) = 1 + x*C(x)^2.

a(n) = Sum_{k=0..n} (2*k+1) * binomial(4*k+1,k) * binomial(4*n-2*k+1,n-k)/((4*k+1) * (4*n-2*k+1)).

Prog

(PARI)
lista(nn, k=2, p=4, s=2, 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 A396447.

Sequence in context: A151418 A154273 A381783 * A253256 A163203 A142722

Adjacent sequences: A396430 A396431 A396432 * A396434 A396435 A396436

Keyword

nonn

Author

Seiichi Manyama, May 25 2026

Status

approved