A396711 Square array A(n,k), n>=1, k>=0, read by antidiagonals downwards, where A(n,k) = n! * [x^n] -W_k(-x)/(1 + W_k(-x)) and W_k(x) is the k-th iterate of LambertW(x).

1, 1, 2, 1, 4, 6, 1, 6, 27, 24, 1, 8, 60, 256, 120, 1, 10, 105, 836, 3125, 720, 1, 12, 162, 1908, 14920, 46656, 5040, 1, 14, 231, 3616, 44265, 324582, 823543, 40320, 1, 16, 312, 6104, 102800, 1249398, 8329552, 16777216, 362880, 1, 18, 405, 9516, 205045, 3550644, 41546757, 246318088, 387420489, 3628800

Offset

1,3

Links

Table of n, a(n) for n=1..55.

Formula

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

A(n,k) = (n-1)! * Sum_{x_1, x_2, ..., x_{k+1} >= 0 and x_1 + x_2 + ... + x_{k+1} = n-1} (x_{k+1} + 1) * Product_{i=1..k} ((n - Sum_{j=1..i-1} x_j)^(x_i) / x_i!).

Example

Square array begins:
    1,     1,      1,       1,       1,       1, ...
    2,     4,      6,       8,      10,      12, ...
    6,    27,     60,     105,     162,     231, ...
   24,   256,    836,    1908,    3616,    6104, ...
  120,  3125,  14920,   44265,  102800,  205045, ...
  720, 46656, 324582, 1249398, 3550644, 8358900, ...
  ...

Prog

(PARI) a(n, k) = if(k==0, n!, sum(j=1, n, n^(n-j)*binomial(n-1, j-1)*a(j, k-1)));

Crossrefs

Columns k=0..3 give A000142, A000312, A396712, A396713.

Cf. A396676.

Sequence in context: A098473 A121757 A391886 * A219441 A219142 A220226

Adjacent sequences: A396705 A396706 A396707 * A396712 A396713 A396714

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 03 2026

Status

approved