A346519 - OEIS

%I #16 Apr 07 2022 10:52:25

%S 1,2,9,59,442,3799,36332,379831,4288933,51867573,667168482,9076862555,

%T 130018298663,1953284957029,30675458303547,502166867458649,

%U 8547908294767932,150965367603029126,2760941474553823577,52196915577464262360,1018499212583077293854

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

%C 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.

%H Alois P. Heinz, <a href="/A346519/b346519.txt">Table of n, a(n) for n = 0..510</a>

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

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

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

%e a(0) = 1: {}.

%e a(1) = 2: 01, 0|1.

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

%p g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(

%p      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p s:= proc(n) option remember; expand(`if`(n=0, 1,

%p       x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))

%p     end:

%p S:= proc(n, k) option remember; coeff(s(n), x, k) end:

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p      `if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))

%p     end:

%p a:= n-> add(S(n, j)*b(n, j), j=0..n):

%p seq(a(n), n=0..20);

%t 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];

%t s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];

%t S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, g[n], Sum[b[n - j, i - 1], {j, 0, n}]]];

%t a[n_] := Sum[S[n, j]*b[n, j], {j, 0, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 06 2022, after _Alois P. Heinz_ *)

%Y Main diagonal of A346520.

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

%K nonn

%O 0,2

%A _Alois P. Heinz_, Jul 21 2021