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%I #22 Mar 12 2022 11:33:21
%S 1,2,6,17,53,180,683,2866,13219,66307,358532,2074229,12761831,
%T 83086064,570017222,4106269668,30965072776,243778358992,1998878586251,
%U 17034471643814,150591119435358,1378657063570498,13050460812585580,127553991370245410,1285578058726241427
%N Total number of partitions of all n-multisets {0,...,0,1,2,...,j} for 0 <= j <= n.
%C 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.
%H Alois P. Heinz, <a href="/A346428/b346428.txt">Table of n, a(n) for n = 0..576</a>
%F a(n) = Sum_{j=0..n} A346426(n-j,j).
%e a(2) = 6: 00, 0|0, 01, 0|1, 12, 1|2.
%e 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.
%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(add(S(n-i, j)*b(i, j), j=0..n-i), i=0..n):
%p seq(a(n), n=0..25);
%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[Sum[S[n - i, j]*b[i, j], {j, 0, n - i}], {i, 0, n}];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 12 2022, after _Alois P. Heinz_ *)
%Y Antidiagonal sums of A346426.
%Y Cf. A346490, A346521.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Jul 16 2021
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