A396504 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).

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, 54144, 468512, 0, 1, 12, 260, 6608, 156240, 2730784, 23762752, 0, 1, 14, 360, 11320, 355968, 9468096, 172930816, 1458000000, 0, 1, 16, 476, 17832, 700560, 25170368, 716663232, 13214614144, 105046700288, 0

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Formula

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.

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).

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.

Example

Square array begins:
  1,      1,       1,       1,        1,        1, ...
  0,      2,       4,       6,        8,       10, ...
  0,     20,      56,     108,      176,      260, ...
  0,    392,    1432,    3408,     6608,    11320, ...
  0,  11664,   54144,  156240,   355968,   700560, ...
  0, 468512, 2730784, 9468096, 25170368, 56596960, ...
  ...

Prog

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

Crossrefs

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

Cf. A396430, A396502.

Sequence in context: A378238 A396503 A378240 * A364228 A112899 A212808

Adjacent sequences: A396501 A396502 A396503 * A396505 A396506 A396507

Keyword

nonn,tabl

Author

Seiichi Manyama, May 28 2026

Status

approved