A396433 - OEIS

%I #21 May 27 2026 11:36:21

%S 1,2,11,79,645,5688,52850,510147,5070522,51581908,534750083,

%T 5631311687,60088612174,648403965361,7064553448007,77615909028006,

%U 858980155363199,9567496484998948,107170256222238701,1206531521792903644,13644565395364715036,154931191054173346846

%N 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.

%H Seiichi Manyama, <a href="/A396433/b396433.txt">Table of n, a(n) for n = 0..924</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>

%F 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.

%F 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)).

%o (PARI)

%o lista(nn, k=2, p=4, s=2, r=1) = {

%o   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))));

%o   my(TK=T^k);

%o   TK[1, ];

%o };

%Y Column k=2 of A396447.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 25 2026