A346490 - OEIS

%I #18 Mar 12 2022 11:33:30

%S 1,2,6,18,61,228,926,4126,19688,101582,556763,3258810,20134527,

%T 131591030,902915694,6506096000,48986713992,385159376478,

%U 3151457714098,26806601933838,236457090358459,2160451562170100,20408176433186475,199086685731569740,2002713693735431017

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

%C Also total number of factorizations of Product_{i=1..n-j} prime(i) * Product_{i=1..j} prime(i) for 0 <= j <= n.

%H Alois P. Heinz, <a href="/A346490/b346490.txt">Table of n, a(n) for n = 0..576</a>

%F a(n) = Sum_{j=0..n} A001055(A002110(n-j)*A002110(j)).

%F a(n) = Sum_{j=0..n} A346500(n-j,j).

%p b:= proc(n) option remember; `if`(n=0, 1,

%p       add(b(n-j)*binomial(n-1, j-1), j=1..n))

%p     end:

%p A:= proc(n, k) option remember; `if`(n<k, A(k, n),

%p      `if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)

%p       *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))

%p     end:

%p a:= n-> add(A(n-j, j), j=0..n):

%p seq(a(n), n=0..24);

%t b[n_] := b[n] = If[n == 0, 1,

%t      Sum[b[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]];

%t A[n_, k_] := A[n, k] = If[n < k, A[k, n],

%t      If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]

%t      *Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];

%t a[n_] := Sum[A[n - j, j], {j, 0, n}];

%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Mar 12 2022, after _Alois P. Heinz_ *)

%Y Antidiagonal sums of A346500.

%Y Cf. A001055, A002110, A346428, A346518.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Jul 19 2021