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A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

%I A330463 #18 Feb 22 2023 09:58:36
%S A330463 1,0,1,0,2,0,0,3,2,0,0,5,4,0,0,0,7,11,1,0,0,0,11,20,6,0,0,0,0,15,40,
%T A330463 16,0,0,0,0,0,22,68,40,3,0,0,0,0,0,30,120,91,11,0,0,0,0,0,0,42,195,
%U A330463 186,41,0,0,0,0,0,0,0,56,320,367,105,3,0,0,0,0,0,0
%N A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.
%H A330463 Andrew Howroyd, <a href="/A330463/b330463.txt">Table of n, a(n) for n = 0..1325</a> (rows n = 0..50)
%F A330463 G.f.: Product_{j>=1} (1 + y*x^j)^A000041(j). - _Andrew Howroyd_, Dec 29 2019
%e A330463 Triangle begins:
%e A330463   1
%e A330463   0  1
%e A330463   0  2  0
%e A330463   0  3  2  0
%e A330463   0  5  4  0  0
%e A330463   0  7 11  1  0  0
%e A330463   0 11 20  6  0  0  0
%e A330463   0 15 40 16  0  0  0  0
%e A330463   0 22 68 40  3  0  0  0  0
%e A330463   ...
%e A330463 Row n = 5 counts the following sets of multisets:
%e A330463   {{5}}          {{1},{4}}        {{1},{2},{1,1}}
%e A330463   {{1,4}}        {{2},{3}}
%e A330463   {{2,3}}        {{1},{1,3}}
%e A330463   {{1,1,3}}      {{1},{2,2}}
%e A330463   {{1,2,2}}      {{2},{1,2}}
%e A330463   {{1,1,1,2}}    {{3},{1,1}}
%e A330463   {{1,1,1,1,1}}  {{1},{1,1,2}}
%e A330463                  {{1,1},{1,2}}
%e A330463                  {{2},{1,1,1}}
%e A330463                  {{1},{1,1,1,1}}
%e A330463                  {{1,1},{1,1,1}}
%p A330463 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
%p A330463        combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
%p A330463     end:
%p A330463 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A330463 seq(T(n), n=0..14);  # _Alois P. Heinz_, Dec 30 2019
%t A330463 ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
%t A330463 Table[Length[Select[ppl[n,2],And[UnsameQ@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
%t A330463 (* Second program: *)
%t A330463 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
%t A330463      PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
%t A330463 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
%t A330463 T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, May 18 2021, after _Alois P. Heinz_ *)
%o A330463 (PARI)
%o A330463 A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, Vecrev(v[n],n))}
%o A330463 {my(T=A(12)); for(n=1, #T, print(T[n]))} \\ _Andrew Howroyd_, Dec 29 2019
%Y A330463 Row sums are A261049.
%Y A330463 Column k = 1 is A000041.
%Y A330463 Multisets of multisets are A061260, with row sums A001970.
%Y A330463 Sets of sets are A330462, with row sums A050342.
%Y A330463 Multisets of sets are A285229, with row sums A089259.
%Y A330463 Sets of disjoint sets are A330460, with row sums A294617.
%Y A330463 Cf. A007716, A060016, A063834, A255906, A271619, A279785, A360742.
%K A330463 nonn,tabl
%O A330463 0,5
%A A330463 _Gus Wiseman_, Dec 19 2019