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A346518 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} into distinct multisets for 0 <= j <= n.

%I A346518 #11 Apr 07 2022 10:52:14
%S A346518 1,2,5,16,53,202,826,3724,17939,93390,516125,3042412,18923139,
%T A346518 124368810,857827458,6208594458,46937360868,370335617694,
%U A346518 3039823038753,25928519847988,229285625745624,2099543718917418,19872430464012935,194203934113959970,1956736801957704866
%N A346518 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} into distinct multisets for 0 <= j <= n.
%C A346518 Also total number of factorizations of Product_{i=1..n-j} prime(i) * Product_{i=1..j} prime(i) into distinct factors for 0 <= j <= n.
%H A346518 Alois P. Heinz, <a href="/A346518/b346518.txt">Table of n, a(n) for n = 0..576</a>
%F A346518 a(n) = Sum_{j=0..n} A045778(A002110(n-j)*A002110(j)).
%F A346518 a(n) = Sum_{j=0..n} A346517(n-j,j).
%p A346518 b:= proc(n) option remember; `if`(n=0, 1,
%p A346518       add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A346518     end:
%p A346518 A:= proc(n, k) option remember; `if`(n<k, A(k, n),
%p A346518      `if`(k=0, b(n), (A(n+1, k-1)-add(A(n-k+j, j)
%p A346518       *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
%p A346518     end:
%p A346518 a:= n-> add(A(n-j, j), j=0..n):
%p A346518 seq(a(n), n=0..24);
%t A346518 (* Q is A322770 *)
%t A346518 Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2) (Q[m + 2, n - 1] +
%t A346518      Q[m + 1, n - 1] - Sum[Binomial[n - 1, k] Q[m, k], {k, 0, n - 1}])];
%t A346518 A[n_, k_] := Q[Abs[n - k], Min[n, k]];
%t A346518 a[n_] := Sum[A[n - j, j], {j, 0, n}];
%t A346518 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Apr 06 2022 *)
%Y A346518 Antidiagonal sums of A346517.
%Y A346518 Cf. A002110, A045778, A346490.
%K A346518 nonn
%O A346518 0,2
%A A346518 _Alois P. Heinz_, Jul 21 2021