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A284992 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j^k) in powers of x.

%I A284992 #29 Feb 10 2021 03:41:16
%S A284992 1,1,1,1,1,1,1,1,2,2,1,1,4,5,2,1,1,8,13,8,3,1,1,16,35,31,16,4,1,1,32,
%T A284992 97,119,83,28,5,1,1,64,275,457,433,201,49,6,1,1,128,793,1763,2297,
%U A284992 1476,487,83,8,1,1,256,2315,6841,12421,11113,4962,1141,142,10,1,1
%N A284992 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j^k) in powers of x.
%H A284992 Seiichi Manyama, <a href="/A284992/b284992.txt">Antidiagonals n = 0..139, flattened</a>
%F A284992 G.f. of column k: Product_{j>=1} (1+x^j)^(j^k).
%e A284992 Square array begins:
%e A284992   1,  1,   1,    1,     1,      1,       1,        1, ...
%e A284992   1,  1,   1,    1,     1,      1,       1,        1, ...
%e A284992   1,  2,   4,    8,    16,     32,      64,      128, ...
%e A284992   2,  5,  13,   35,    97,    275,     793,     2315, ...
%e A284992   2,  8,  31,  119,   457,   1763,    6841,    26699, ...
%e A284992   3, 16,  83,  433,  2297,  12421,   68393,   382573, ...
%e A284992   4, 28, 201, 1476, 11113,  85808,  678101,  5466916, ...
%e A284992   5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...
%p A284992 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A284992       add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))
%p A284992     end:
%p A284992 A:= (n, k)-> b(n$2, k):
%p A284992 seq(seq(A(n, d-n), n=0..d), d=0..14);  # _Alois P. Heinz_, Oct 16 2017
%t A284992 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
%t A284992      Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]];
%t A284992 A[n_, k_] := b[n, n, k];
%t A284992 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 10 2021, after _Alois P. Heinz_ *)
%Y A284992 Columns k=0-5 give A000009, A026007, A027998, A248882, A248883, A248884.
%Y A284992 Rows (0+1),2-3 give: A000012, A000079, A007689.
%Y A284992 Main diagonal gives A270917.
%Y A284992 Cf. A283272, A284993.
%K A284992 nonn,tabl
%O A284992 0,9
%A A284992 _Seiichi Manyama_, Apr 07 2017