A396434 - OEIS

%I #22 May 27 2026 11:35:24

%S 1,2,12,94,836,8025,81136,852012,9210604,101884076,1148162064,

%T 13138949028,152296165996,1784568913542,21106289095896,

%U 251636459582520,3021119305369280,36494117062138956,443229504130411376,5409100104838981832,66296660011662002352,815719349332409935946

%N 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.

%H Seiichi Manyama, <a href="/A396434/b396434.txt">Table of n, a(n) for n = 0..895</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) ))^(1/3), where H_k(x) is the k-th iteration of x*(1 - x)^3.

%F 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)).

%o (PARI)

%o lista(nn, k=2, p=4, s=3, 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 A396448.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 25 2026