A396504 - OEIS

%I #28 May 30 2026 17:02:25

%S 1,1,0,1,2,0,1,4,20,0,1,6,56,392,0,1,8,108,1432,11664,0,1,10,176,3408,

%T 54144,468512,0,1,12,260,6608,156240,2730784,23762752,0,1,14,360,

%U 11320,355968,9468096,172930816,1458000000,0,1,16,476,17832,700560,25170368,716663232,13214614144,105046700288,0

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

%F E.g.f. of column k: (1/x) * Series_Reversion( H_k(x) ), where H_k(x) is the k-th iterate of U(x)*exp(-4*U(x)) and U(x) = -LambertW(-2*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) * (4*x_{j+1} - 2*x_j + 2)^(x_{j+1} - x_j - 1) * binomial(x_{j+1},x_j).

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

%e Square array begins:

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

%e   0,      2,       4,       6,        8,       10, ...

%e   0,     20,      56,     108,      176,      260, ...

%e   0,    392,    1432,    3408,     6608,    11320, ...

%e   0,  11664,   54144,  156240,   355968,   700560, ...

%e   0, 468512, 2730784, 9468096, 25170368, 56596960, ...

%e   ...

%o (PARI)

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

%o   my(T=matrix(n+1, n+1, row, col, my(xr=row-1, xc=col-1); if(xc<xr, 0, (s*xr+r)*(p*xc-(p-s)*xr+r)^(xc-xr-1)*binomial(xc, xr))));

%o   my(TK=T^k);

%o   TK[1, n+1];

%o };

%Y Columns k=0..1 give A000007, A396509.

%Y Cf. A396430, A396502.

%K nonn,tabl

%O 0,5

%A _Seiichi Manyama_, May 28 2026