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

Author

Alois P. Heinz, Jul 21 2021

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Main diagonal of A346520.

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

Programs

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 *)

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).