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%I #26 Apr 07 2022 10:51:55
%S 1,2,11,74,592,5317,52902,572402,6670707,83025806,1096662664,
%T 15292076689,224145880470,3440981816071,55153081768896,
%U 920494136057715,15959177281931953,286834809549486462,5334308665713522860,102476857445135062727,2030589375575413246579
%N Number of partitions of the 2n-multiset {0,...,0,1,2,...,n}.
%C Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1); a(2) = 11: 2*2*3*5, 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 2*2*15, 3*20, 2*30, 60.
%H Alois P. Heinz, <a href="/A346424/b346424.txt">Table of n, a(n) for n = 0..509</a>
%F a(n) = A001055(A000079(n)*A070826(n+1)).
%F a(n) = Sum_{j=0..n} A048993(n,j)*A292508(n,j+1).
%F a(n) = A346426(n,n).
%e a(0) = 1: {}.
%e a(1) = 2: 01, 0|1.
%e a(2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
%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, `if`(i=0,
%p combinat[numbpart](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..21);
%t s[n_] := s[n] = Expand[If[n == 0, 1,
%t 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,
%t PartitionsP[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, 21}] (* _Jean-François Alcover_, Apr 06 2022, after _Alois P. Heinz_ *)
%Y Main diagonal of A346426.
%Y Cf. A000040, A000041, A000079, A000110, A001055, A048993, A070826, A292508, A346519.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Jul 16 2021
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