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

1, 2, 6, 18, 61, 228, 926, 4126, 19688, 101582, 556763, 3258810, 20134527, 131591030, 902915694, 6506096000, 48986713992, 385159376478, 3151457714098, 26806601933838, 236457090358459, 2160451562170100, 20408176433186475, 199086685731569740, 2002713693735431017

Offset

0,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..576

Formula

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

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

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n<k, A(k, n),
     `if`(k=0, b(n), (A(n+1, k-1)+add(A(n-k+j, j)
      *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
    end:
a:= n-> add(A(n-j, j), j=0..n):
seq(a(n), n=0..24);

Mathematica

b[n_] := b[n] = If[n == 0, 1,
     Sum[b[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]];
A[n_, k_] := A[n, k] = If[n < k, A[k, n],
     If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]
     *Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)

Crossrefs

Antidiagonal sums of A346500.

Cf. A001055, A002110, A346428, A346518.

Sequence in context: A148462 A123639 A228448 * A177473 A177471 A303117

Adjacent sequences: A346487 A346488 A346489 * A346491 A346492 A346493

Keyword

nonn

Author

Alois P. Heinz, Jul 19 2021

Status

approved