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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).
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%I #25 Sep 07 2023 15:50:28

%S 1,1,1,1,1,1,1,1,2,2,1,1,4,5,2,1,1,8,13,7,3,1,1,16,35,25,15,4,1,1,32,

%T 97,91,77,25,5,1,1,64,275,337,405,161,43,6,1,1,128,793,1267,2177,1069,

%U 393,64,8,1,1,256,2315,4825,11925,7313,3799,726,120,10

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

%H Alois P. Heinz, <a href="/A292189/b292189.txt">Rows n = 0..150, flattened</a>

%e Square array begins:

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

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

%e 1, 2, 4, 8, 16, ...

%e 2, 5, 13, 35, 97, ...

%e 2, 7, 25, 91, 337, ...

%p b:= proc(n, i, k) option remember; (m->

%p `if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+

%p `if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)

%p end:

%p A:= (n, k)-> b(n$2, k):

%p seq(seq(A(n, d-n), n=0..d), d=0..14); # _Alois P. Heinz_, Sep 11 2017

%t m = 14;

%t col[k_] := col[k] = Product[1 + j^k*x^j, {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;

%t A[n_, k_] := col[k][[n+1]];

%t Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 11 2021 *)

%Y Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.

%Y Rows 0+1, 2, 3 give A000012, A000079, A007689.

%Y Main diagonal gives A292190.

%Y Cf. A292166.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Sep 11 2017