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A346428 Total number of partitions of all n-multisets {0,...,0,1,2,...,j} for 0 <= j <= n.

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