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A346521 Total number of partitions of all n-multisets {0,...,0,1,2,...,j} into distinct multisets for 0 <= j <= n.

%I A346521 #18 Jul 31 2021 09:43:41
%S A346521 1,2,5,15,46,161,624,2669,12483,63261,344631,2005058,12390086,
%T A346521 80945545,556896913,4021109557,30382294412,239589006143,1967343509525,
%U A346521 16786587081641,148561276135546,1361378815644787,12897870827339021,126158299918183469,1272377007364596242
%N A346521 Total number of partitions of all n-multisets {0,...,0,1,2,...,j} into distinct multisets for 0 <= j <= n.
%C A346521 Also total number of factorizations of 2^(n-j) * Product_{i=1..j} prime(i+1) into distinct factors for 0 <= j <= n; a(2) = 5: 4, 2*3, 6, 3*5, 15; a(3) = 15: 2*4, 8, 3*4, 2*6, 12, 2*3*5, 5*6, 3*10, 2*15, 30, 3*5*7, 7*15, 5*21, 3*35, 105.
%H A346521 Alois P. Heinz, <a href="/A346521/b346521.txt">Table of n, a(n) for n = 0..576</a>
%F A346521 a(n) = Sum_{j=0..n} A346520(n-j,j).
%F A346521 a(n) = Sum_{j=0..n} A045778(A000079(n-j)*A070826(j+1)).
%e A346521 a(2) = 5: 00, 01, 0|1, 12, 1|2.
%e A346521 a(3) = 15: 000, 0|00, 001, 00|1, 0|01, 012, 0|12, 02|1, 01|2, 0|1|2, 123, 1|23, 13|2, 12|3, 1|2|3.
%p A346521 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A346521      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A346521     end:
%p A346521 s:= proc(n) option remember; expand(`if`(n=0, 1,
%p A346521       x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
%p A346521     end:
%p A346521 S:= proc(n, k) option remember; coeff(s(n), x, k) end:
%p A346521 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A346521      `if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))
%p A346521     end:
%p A346521 A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
%p A346521 a:= n-> add(A(n-j, j), j=0..n):
%p A346521 seq(a(n), n=0..24);
%t A346521 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 A346521 s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
%t A346521 S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
%t A346521 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 A346521 A[n_, k_] := Sum[S[k, j]*b[n, j], {j, 0, k}];
%t A346521 a[n_] := Sum[A[n - j, j], {j, 0, n}];
%t A346521 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Jul 31 2021, after _Alois P. Heinz_ *)
%Y A346521 Antidiagonal sums of A346520.
%Y A346521 Cf. A000009, A000040, A000079, A000110, A045778, A070826, A346428.
%K A346521 nonn
%O A346521 0,2
%A A346521 _Alois P. Heinz_, Jul 21 2021