A396711 - OEIS

%I #15 Jun 04 2026 00:44:57

%S 1,1,2,1,4,6,1,6,27,24,1,8,60,256,120,1,10,105,836,3125,720,1,12,162,

%T 1908,14920,46656,5040,1,14,231,3616,44265,324582,823543,40320,1,16,

%U 312,6104,102800,1249398,8329552,16777216,362880,1,18,405,9516,205045,3550644,41546757,246318088,387420489,3628800

%N 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).

%F 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.

%F 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!).

%e Square array begins:

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

%e     2,     4,      6,       8,      10,      12, ...

%e     6,    27,     60,     105,     162,     231, ...

%e    24,   256,    836,    1908,    3616,    6104, ...

%e   120,  3125,  14920,   44265,  102800,  205045, ...

%e   720, 46656, 324582, 1249398, 3550644, 8358900, ...

%e   ...

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

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

%Y Cf. A396676.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, Jun 03 2026