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

1, 2, 5, 16, 53, 202, 826, 3724, 17939, 93390, 516125, 3042412, 18923139, 124368810, 857827458, 6208594458, 46937360868, 370335617694, 3039823038753, 25928519847988, 229285625745624, 2099543718917418, 19872430464012935, 194203934113959970, 1956736801957704866

Offset

0,2

Comments

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.

Links

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

Formula

a(n) = Sum_{j=0..n} A045778(A002110(n-j)*A002110(j)).

a(n) = Sum_{j=0..n} A346517(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

(* Q is A322770 *)
Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2) (Q[m + 2, n - 1] +
     Q[m + 1, n - 1] - Sum[Binomial[n - 1, k] Q[m, k], {k, 0, n - 1}])];
A[n_, k_] := Q[Abs[n - k], Min[n, k]];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 06 2022 *)

Crossrefs

Antidiagonal sums of A346517.

Cf. A002110, A045778, A346490.

Sequence in context: A108521 A148397 A148398 * A100442 A081126 A018191

Adjacent sequences: A346515 A346516 A346517 * A346519 A346520 A346521

Keyword

nonn

Author

Alois P. Heinz, Jul 21 2021

Status

approved