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A283674 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-x^j)^(j^(k*j)) in powers of x.

%I A283674 #30 Mar 17 2017 10:09:09
%S A283674 1,1,1,1,1,2,1,1,5,3,1,1,17,32,5,1,1,65,746,298,7,1,1,257,19748,66418,
%T A283674 3531,11,1,1,1025,531698,16799044,9843707,51609,15,1,1,4097,14349932,
%U A283674 4295531890,30535636881,2187941520,894834,22,1,1,16385,387424586,1099526502508,95371863221411,101591759812967,680615139257,17980052,30
%N A283674 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-x^j)^(j^(k*j)) in powers of x.
%H A283674 Alois P. Heinz, <a href="/A283674/b283674.txt">Antidiagonals n = 0..52</a>
%F A283674 G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j^(k*j)).
%e A283674 Square array begins:
%e A283674    1,   1,     1,        1, ...
%e A283674    1,   1,     1,        1, ...
%e A283674    2,   5,    17,       65, ...
%e A283674    3,  32,   746,    19748, ...
%e A283674    5, 298, 66418, 16799044, ...
%p A283674 with(numtheory):
%p A283674 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
%p A283674       d*d^(k*d), d=divisors(j))*A(n-j, k), j=1..n)/n)
%p A283674     end:
%p A283674 seq(seq(A(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Mar 15 2017
%t A283674 A[n_, k_] := If[n==0, 1, Sum[Sum[d*d^(k*d), {d, Divisors[j]}] *A[n - j, k], {j, n}] / n]; Flatten[Table[A[d - n,  n],{d, 0, 10},{n, d, 0, -1}]] (* _Indranil Ghosh_, Mar 17 2017 *)
%o A283674 (PARI) A(n, k) = if(n==0, 1, sum(j=1, n, sumdiv(j, d, d*d^(k*d)) * A(n - j, k))/n);
%o A283674 {for(d=0, 10, for(n=0, d, print1(A(n, d - n),", ");); print(););} \\ _Indranil Ghosh_, Mar 17 2017
%Y A283674 Columns k=0-4 give A000041, A023880, A283579, A283580, A283510.
%Y A283674 Rows give: 0-1: A000012, 2: A052539, 3: A283716.
%Y A283674 Main diagonal gives A283719.
%Y A283674 Cf. A283675.
%K A283674 nonn,tabl
%O A283674 0,6
%A A283674 _Seiichi Manyama_, Mar 14 2017