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A346490 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} for 0 <= j <= n.

%I A346490 #18 Mar 12 2022 11:33:30
%S A346490 1,2,6,18,61,228,926,4126,19688,101582,556763,3258810,20134527,
%T A346490 131591030,902915694,6506096000,48986713992,385159376478,
%U A346490 3151457714098,26806601933838,236457090358459,2160451562170100,20408176433186475,199086685731569740,2002713693735431017
%N A346490 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} for 0 <= j <= n.
%C A346490 Also total number of factorizations of Product_{i=1..n-j} prime(i) * Product_{i=1..j} prime(i) for 0 <= j <= n.
%H A346490 Alois P. Heinz, <a href="/A346490/b346490.txt">Table of n, a(n) for n = 0..576</a>
%F A346490 a(n) = Sum_{j=0..n} A001055(A002110(n-j)*A002110(j)).
%F A346490 a(n) = Sum_{j=0..n} A346500(n-j,j).
%p A346490 b:= proc(n) option remember; `if`(n=0, 1,
%p A346490       add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A346490     end:
%p A346490 A:= proc(n, k) option remember; `if`(n<k, A(k, n),
%p A346490      `if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)
%p A346490       *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
%p A346490     end:
%p A346490 a:= n-> add(A(n-j, j), j=0..n):
%p A346490 seq(a(n), n=0..24);
%t A346490 b[n_] := b[n] = If[n == 0, 1,
%t A346490      Sum[b[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]];
%t A346490 A[n_, k_] := A[n, k] = If[n < k, A[k, n],
%t A346490      If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]
%t A346490      *Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
%t A346490 a[n_] := Sum[A[n - j, j], {j, 0, n}];
%t A346490 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Mar 12 2022, after _Alois P. Heinz_ *)
%Y A346490 Antidiagonal sums of A346500.
%Y A346490 Cf. A001055, A002110, A346428, A346518.
%K A346490 nonn
%O A346490 0,2
%A A346490 _Alois P. Heinz_, Jul 19 2021