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A346519 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n} into distinct multisets.

%I A346519 #16 Apr 07 2022 10:52:25
%S A346519 1,2,9,59,442,3799,36332,379831,4288933,51867573,667168482,9076862555,
%T A346519 130018298663,1953284957029,30675458303547,502166867458649,
%U A346519 8547908294767932,150965367603029126,2760941474553823577,52196915577464262360,1018499212583077293854
%N A346519 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n} into distinct multisets.
%C A346519 Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1) into distinct factors; a(2) = 9: 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 3*20, 2*30, 60.
%H A346519 Alois P. Heinz, <a href="/A346519/b346519.txt">Table of n, a(n) for n = 0..510</a>
%F A346519 a(n) = A045778(A000079(n)*A070826(n+1)).
%F A346519 a(n) = Sum_{j=0..n} Stirling2(n,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).
%F A346519 a(n) = A346520(n,n).
%e A346519 a(0) = 1: {}.
%e A346519 a(1) = 2: 01, 0|1.
%e A346519 a(2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|12, 0|012, 0012.
%p A346519 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A346519      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A346519     end:
%p A346519 s:= proc(n) option remember; expand(`if`(n=0, 1,
%p A346519       x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
%p A346519     end:
%p A346519 S:= proc(n, k) option remember; coeff(s(n), x, k) end:
%p A346519 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A346519      `if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))
%p A346519     end:
%p A346519 a:= n-> add(S(n, j)*b(n, j), j=0..n):
%p A346519 seq(a(n), n=0..20);
%t A346519 g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
%t A346519 s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
%t A346519 S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
%t A346519 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, g[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
%t A346519 a[n_] := Sum[S[n, j]*b[n, j], {j, 0, n}];
%t A346519 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 06 2022, after _Alois P. Heinz_ *)
%Y A346519 Main diagonal of A346520.
%Y A346519 Cf. A000009, A000040, A000079, A045778, A048993, A070826, A346424.
%K A346519 nonn
%O A346519 0,2
%A A346519 _Alois P. Heinz_, Jul 21 2021