A346519 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n} into distinct multisets.

1, 2, 9, 59, 442, 3799, 36332, 379831, 4288933, 51867573, 667168482, 9076862555, 130018298663, 1953284957029, 30675458303547, 502166867458649, 8547908294767932, 150965367603029126, 2760941474553823577, 52196915577464262360, 1018499212583077293854

Offset

0,2

Comments

Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1) into distinct factors; a(2) = 9: 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 3*20, 2*30, 60.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..510

Formula

a(n) = A045778(A000079(n)*A070826(n+1)).

a(n) = Sum_{j=0..n} Stirling2(n,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).

a(n) = A346520(n,n).

Example

a(0) = 1: {}.
a(1) = 2: 01, 0|1.
a(2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|12, 0|012, 0012.

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
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, g(n), add(b(n-j, i-1), j=0..n)))
    end:
a:= n-> add(S(n, j)*b(n, j), j=0..n):
seq(a(n), n=0..20);

Mathematica

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
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, g[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
a[n_] := Sum[S[n, j]*b[n, j], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 06 2022, after Alois P. Heinz *)

Crossrefs

Main diagonal of A346520.

Cf. A000009, A000040, A000079, A045778, A048993, A070826, A346424.

Sequence in context: A005364 A267464 A184355 * A151616 A199543 A009636

Adjacent sequences: A346516 A346517 A346518 * A346520 A346521 A346522

Keyword

nonn

Author

Alois P. Heinz, Jul 21 2021

Status

approved