A346428 Total number of partitions of all n-multisets {0,...,0,1,2,...,j} for 0 <= j <= n.

1, 2, 6, 17, 53, 180, 683, 2866, 13219, 66307, 358532, 2074229, 12761831, 83086064, 570017222, 4106269668, 30965072776, 243778358992, 1998878586251, 17034471643814, 150591119435358, 1378657063570498, 13050460812585580, 127553991370245410, 1285578058726241427

Offset

0,2

Comments

Also total number of factorizations of 2^(n-j) * Product_{i=1..j} prime(i+1) for 0 <= j <= n; a(2) = 6: 2*2, 4, 2*3, 6, 3*5, 15; a(3) = 17: 2*2*2, 2*4, 8, 2*2*3, 3*4, 2*6, 12, 2*3*5, 5*6, 3*10, 2*15, 30, 3*5*7, 7*15, 5*21, 3*35, 105.

Links

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

Formula

a(n) = Sum_{j=0..n} A346426(n-j,j).

Example

a(2) = 6: 00, 0|0, 01, 0|1, 12, 1|2.
a(3) = 17: 000, 0|00, 0|0|0, 001, 00|1, 0|01, 0|0|1, 012, 0|12, 02|1, 01|2, 0|1|2, 123, 1|23, 13|2, 12|3, 1|2|3.

Maple

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
a:= n-> add(add(S(n-i, j)*b(i, j), j=0..n-i), i=0..n):
seq(a(n), n=0..25);

Mathematica

s[n_] := s[n] = Expand[If[n == 0, 1,
     x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0,
     PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
a[n_] := Sum[Sum[S[n - i, j]*b[i, j], {j, 0, n - i}], {i, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)

Crossrefs

Antidiagonal sums of A346426.

Cf. A346490, A346521.

Sequence in context: A364329 A367056 A368598 * A148453 A097514 A108630

Adjacent sequences: A346425 A346426 A346427 * A346429 A346430 A346431

Keyword

nonn

Author

Alois P. Heinz, Jul 16 2021

Status

approved