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.

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, 1, 6, 50, 358, 1850, 5688, 7084, 0, 1, 7, 69, 610, 4195, 19757, 52850, 53820, 0, 1, 8, 91, 957, 8225, 52526, 221598, 510147, 420732, 0, 1, 9, 116, 1414, 14590, 118040, 688703, 2577350, 5070522, 3362260, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Wikipedia, Fuss-Catalan number

Formula

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.

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).

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.

Example

Square array begins:
  1,    1,     1,      1,      1,       1,       1, ...
  0,    1,     2,      3,      4,       5,       6, ...
  0,    4,    11,     21,     34,      50,      69, ...
  0,   22,    79,    186,    358,     610,     957, ...
  0,  140,   645,   1850,   4195,    8225,   14590, ...
  0,  969,  5688,  19757,  52526,  118040,  235984, ...
  0, 7084, 52850, 221598, 688703, 1769425, 3978114, ...
  ...

Prog

(PARI)
a(n, k, p=4, s=2, r=1) = {
  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))));
  my(TK=T^k);
  TK[1, n+1];
};

Crossrefs

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

Cf. A396430.

Sequence in context: A309148 A396413 A396993 * A397000 A396972 A351761

Adjacent sequences: A396444 A396445 A396446 * A396448 A396449 A396450

Keyword

nonn,tabl

Author

Seiichi Manyama, May 26 2026

Status

approved