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A292189 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).

%I A292189 #25 Sep 07 2023 15:50:28
%S A292189 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 A292189 97,91,77,25,5,1,1,64,275,337,405,161,43,6,1,1,128,793,1267,2177,1069,
%U A292189 393,64,8,1,1,256,2315,4825,11925,7313,3799,726,120,10
%N A292189 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 A292189 Alois P. Heinz, <a href="/A292189/b292189.txt">Rows n = 0..150, flattened</a>
%e A292189 Square array begins:
%e A292189    1, 1,  1,  1,   1, ...
%e A292189    1, 1,  1,  1,   1, ...
%e A292189    1, 2,  4,  8,  16, ...
%e A292189    2, 5, 13, 35,  97, ...
%e A292189    2, 7, 25, 91, 337, ...
%p A292189 b:= proc(n, i, k) option remember; (m->
%p A292189       `if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
%p A292189       `if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
%p A292189     end:
%p A292189 A:= (n, k)-> b(n$2, k):
%p A292189 seq(seq(A(n, d-n), n=0..d), d=0..14);  # _Alois P. Heinz_, Sep 11 2017
%t A292189 m = 14;
%t A292189 col[k_] := col[k] = Product[1 + j^k*x^j, {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
%t A292189 A[n_, k_] := col[k][[n+1]];
%t A292189 Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 11 2021 *)
%Y A292189 Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.
%Y A292189 Rows 0+1, 2, 3 give A000012, A000079, A007689.
%Y A292189 Main diagonal gives A292190.
%Y A292189 Cf. A292166.
%K A292189 nonn,tabl
%O A292189 0,9
%A A292189 _Seiichi Manyama_, Sep 11 2017