A396500 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) with G(x) = exp(x*G(x)^3).

1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 18, 100, 0, 1, 4, 33, 311, 2197, 0, 1, 5, 52, 669, 8096, 65536, 0, 1, 6, 75, 1210, 20121, 283677, 2476099, 0, 1, 7, 102, 1970, 41272, 808653, 12541396, 113379904, 0, 1, 8, 133, 2985, 75125, 1869694, 40852041, 670974607, 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) ), where H_k(x) is the k-th iterate of U(x)*exp(-3*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} (x_j + 1) * (3*x_{j+1} - 2*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} (j+1) * (3*n-2*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,     18,     33,      52,      75, ...
  0,   100,    311,    669,    1210,    1970, ...
  0,  2197,   8096,  20121,   41272,   75125, ...
  0, 65536, 283677, 808653, 1869694, 3785430, ...
  ...

Prog

(PARI)
a(n, k, p=3, s=1, 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..2 give A000007, A052752, A396590.

Cf. A396412, A396499, A396588.

Sequence in context: A384862 A384788 A384752 * A396501 A199292 A152779

Adjacent sequences: A396497 A396498 A396499 * A396501 A396502 A396503

Keyword

nonn,tabl

Author

Seiichi Manyama, May 28 2026

Status

approved