A396447 - OEIS

A396447

Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where A(n,k) = [x^n] (F_k(x)/x)^(1/2) and F_k(x) is the k-th iteration of x*G4(x)^2 with G4(x) = 1 + x*G4(x)^4.

%I #19 May 30 2026 15:52:43

%S 1,1,0,1,1,0,1,2,4,0,1,3,11,22,0,1,4,21,79,140,0,1,5,34,186,645,969,0,

%T 1,6,50,358,1850,5688,7084,0,1,7,69,610,4195,19757,52850,53820,0,1,8,

%U 91,957,8225,52526,221598,510147,420732,0,1,9,116,1414,14590,118040,688703,2577350,5070522,3362260,0

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where A(n,k) = [x^n] (F_k(x)/x)^(1/2) and F_k(x) is the k-th iteration of x*G4(x)^2 with G4(x) = 1 + x*G4(x)^4.

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

%F G.f. of column k: ((1/x) * Series_Reversion( H_k(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,k) = Sum_{0 = x_0 <= x_1 <= ... <= x_{k-1} <= x_k = n} Product_{j=0..k-1} (2*x_j + 1) * binomial(4*x_{j+1} - 2*x_j + 1,x_{j+1} - x_j)/(4*x_{j+1} - 2*x_j + 1).

%F A(n,0) = 0^n; A(n,k) = Sum_{j=0..n} (2*j+1) * binomial(4*n-2*j+1,n-j)/(4*n-2*j+1) * A(j,k-1) for k > 0.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 3, 4, 5, 6, ...

%e 0, 4, 11, 21, 34, 50, 69, ...

%e 0, 22, 79, 186, 358, 610, 957, ...

%e 0, 140, 645, 1850, 4195, 8225, 14590, ...

%e 0, 969, 5688, 19757, 52526, 118040, 235984, ...

%e 0, 7084, 52850, 221598, 688703, 1769425, 3978114, ...

%e ...

%o (PARI)

%o a(n, k, p=4, s=2, r=1) = {

%o my(T=matrix(n+1, n+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, n+1];

%o };

%Y Columns k=0..2 give A000007, A002293, A396433.

%Y Cf. A396430.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, May 26 2026