This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A285229 #37 Feb 22 2023 10:02:40
%S A285229 1,0,1,0,1,1,0,2,1,1,0,2,3,1,1,0,3,4,3,1,1,0,4,8,5,3,1,1,0,5,11,10,5,
%T A285229 3,1,1,0,6,18,16,11,5,3,1,1,0,8,25,29,18,11,5,3,1,1,0,10,38,44,34,19,
%U A285229 11,5,3,1,1,0,12,52,72,55,36,19,11,5,3,1,1
%N A285229 Expansion of g.f. Product_{j>=1} 1/(1-y*x^j)^A000009(j), triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A285229 Alois P. Heinz, <a href="/A285229/b285229.txt">Rows n = 0..200, flattened</a>
%H A285229 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A285229 G.f.: Product_{j>=1} 1/(1-y*x^j)^A000009(j).
%e A285229 T(n,k) is the number of multisets of exactly k partitions of positive integers into distinct parts with total sum of parts equal to n.
%e A285229 T(4,1) = 2: {4}, {31}.
%e A285229 T(4,2) = 3: {3,1}, {21,1}, {2,2}.
%e A285229 T(4,3) = 1: {2,1,1}.
%e A285229 T(4,4) = 1: {1,1,1,1}.
%e A285229 Triangle T(n,k) begins:
%e A285229 1;
%e A285229 0, 1;
%e A285229 0, 1, 1;
%e A285229 0, 2, 1, 1;
%e A285229 0, 2, 3, 1, 1;
%e A285229 0, 3, 4, 3, 1, 1;
%e A285229 0, 4, 8, 5, 3, 1, 1;
%e A285229 0, 5, 11, 10, 5, 3, 1, 1;
%e A285229 0, 6, 18, 16, 11, 5, 3, 1, 1;
%e A285229 0, 8, 25, 29, 18, 11, 5, 3, 1, 1;
%e A285229 0, 10, 38, 44, 34, 19, 11, 5, 3, 1, 1;
%e A285229 0, 12, 52, 72, 55, 36, 19, 11, 5, 3, 1, 1;
%e A285229 0, 15, 75, 110, 96, 60, 37, 19, 11, 5, 3, 1, 1;
%e A285229 ...
%p A285229 with(numtheory):
%p A285229 g:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A285229 `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
%p A285229 end:
%p A285229 b:= proc(n, i) option remember; expand(
%p A285229 `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
%p A285229 x^j*binomial(g(i)+j-1, j), j=0..n/i))))
%p A285229 end:
%p A285229 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A285229 seq(T(n), n=0..16);
%t A285229 L[n_] := QPochhammer[x^2]/QPochhammer[x] + O[x]^n;
%t A285229 A[n_] := Module[{c = L[n]}, CoefficientList[#, y]& /@ CoefficientList[ 1/Product[(1 - x^k*y + O[x]^n)^SeriesCoefficient[c, {x, 0, k}], {k, 1, n}], x]];
%t A285229 A[12] // Flatten (* _Jean-François Alcover_, Jan 19 2020, after _Andrew Howroyd_ *)
%t A285229 g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
%t A285229 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1]*x^j* Binomial[g[i] + j - 1, j], {j, 0, n/i}]]];
%t A285229 T[n_] := CoefficientList[b[n, n] + O[x]^(n+1), x];
%t A285229 T /@ Range[0, 16] // Flatten (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%o A285229 (PARI)
%o A285229 L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
%o A285229 A(n)={my(c=L(n), v=Vec(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}
%o A285229 {my(T=A(12)); for(n=1, #T, print(T[n]))} \\ _Andrew Howroyd_, Dec 29 2019
%Y A285229 Columns k=0..10 give: A000007, A000009 (for n>0), A320787, A320788, A320789, A320790, A320791, A320792, A320793, A320794, A320795.
%Y A285229 Row sums give A089259.
%Y A285229 T(2n,n) give A285230.
%Y A285229 Cf. A061260, A360763.
%K A285229 nonn,tabl
%O A285229 0,8
%A A285229 _Alois P. Heinz_, Apr 14 2017