A396501 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where A(n,k) = n! * [x^n] (F_k(x)/x)^(1/2) and F_k(x) is the k-th iteration of x*G(x)^2 with G(x) = exp(x*G(x)^3).

1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 20, 100, 0, 1, 4, 39, 386, 2197, 0, 1, 5, 64, 948, 11232, 65536, 0, 1, 6, 95, 1876, 34293, 440162, 2476099, 0, 1, 7, 132, 3260, 81088, 1659648, 21779392, 113379904, 0, 1, 8, 175, 5190, 163845, 4665124, 101033859, 1305082130, 6103515625, 0

Offset

0,8

Links

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

Formula

E.g.f. of column k: ((1/x) * Series_Reversion( H_k(x) ))^(1/2), where H_k(x) is the k-th iterate of U(x)*exp(-3*U(x)) and U(x) = -LambertW(-x).

A(n,k) = Sum_{0 = x_0 <= x_1 <= ... <= x_{k-1} <= x_k = n} Product_{j=0..k-1} (2*x_j + 1) * (3*x_{j+1} - x_j + 1)^(x_{j+1} - x_j - 1) * binomial(x_{j+1},x_j).

A(n,0) = 0^n; A(n,k) = Sum_{j=0..n} (2*j+1) * (3*n-j+1)^(n-j-1) * binomial(n,j) * A(j,k-1) for k > 0.

Example

Square array begins:
  1,     1,      1,       1,       1,        1, ...
  0,     1,      2,       3,       4,        5, ...
  0,     7,     20,      39,      64,       95, ...
  0,   100,    386,     948,    1876,     3260, ...
  0,  2197,  11232,   34293,   81088,   163845, ...
  0, 65536, 440162, 1659648, 4665124, 10916480, ...
  ...

Prog

(PARI)
a(n, k, p=3, 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)*(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, A052752.

Cf. A396446, A396503.

Sequence in context: A384788 A384752 A396500 * A199292 A152779 A247373

Adjacent sequences: A396498 A396499 A396500 * A396502 A396503 A396504

Keyword

nonn,tabl

Author

Seiichi Manyama, May 28 2026

Status

approved