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A346520 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A346520 #26 Aug 05 2021 16:12:58
%S A346520 1,1,1,2,2,1,5,5,3,2,15,15,9,5,2,52,52,31,16,7,3,203,203,120,59,25,10,
%T A346520 4,877,877,514,244,100,38,14,5,4140,4140,2407,1112,442,161,56,19,6,
%U A346520 21147,21147,12205,5516,2134,750,249,80,25,8,115975,115975,66491,29505,11147,3799,1213,372,111,33,10
%N A346520 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A346520 Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1) into distinct factors; A(3,1) = 5: 2*3*4, 4*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.
%H A346520 Alois P. Heinz, <a href="/A346520/b346520.txt">Antidiagonals n = 0..140, flattened</a>
%F A346520 A(n,k) = A045778(A000079(n)*A070826(k+1)).
%F A346520 A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).
%e A346520 A(2,2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|12, 0|012, 0012.
%e A346520 Square array A(n,k) begins:
%e A346520   1,  1,   2,   5,   15,    52,   203,    877,    4140, ...
%e A346520   1,  2,   5,  15,   52,   203,   877,   4140,   21147, ...
%e A346520   1,  3,   9,  31,  120,   514,  2407,  12205,   66491, ...
%e A346520   2,  5,  16,  59,  244,  1112,  5516,  29505,  168938, ...
%e A346520   2,  7,  25, 100,  442,  2134, 11147,  62505,  373832, ...
%e A346520   3, 10,  38, 161,  750,  3799, 20739, 121141,  752681, ...
%e A346520   4, 14,  56, 249, 1213,  6404, 36332, 220000, 1413937, ...
%e A346520   5, 19,  80, 372, 1887, 10340, 60727, 379831, 2516880, ...
%e A346520   6, 25, 111, 539, 2840, 16108, 97666, 629346, 4288933, ...
%e A346520   ...
%p A346520 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A346520      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A346520     end:
%p A346520 s:= proc(n) option remember; expand(`if`(n=0, 1,
%p A346520       x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
%p A346520     end:
%p A346520 S:= proc(n, k) option remember; coeff(s(n), x, k) end:
%p A346520 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A346520      `if`(i=0, g(n), add(b(n-j, i-1), j=0..n)))
%p A346520     end:
%p A346520 A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
%p A346520 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A346520 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 A346520 s[n_] := s[n] = Expand[If[n == 0, 1, x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
%t A346520 S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
%t A346520 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 A346520 A[n_, k_] := Sum[S[k, j]*b[n, j], {j, 0, k}];
%t A346520 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jul 31 2021, after _Alois P. Heinz_ *)
%Y A346520 Columns k=0-10 give: A000009, A036469, A346822, A346823, A346824, A346825, A346826, A346827, A346828, A346829, A346830.
%Y A346520 Rows n=0+1,2-10 give: A000110, A087648, A346813, A346814, A346815, A346816, A346817, A346818, A346819, A346820.
%Y A346520 Main diagonal gives A346519.
%Y A346520 Antidiagonal sums give A346521.
%Y A346520 Cf. A000040, A000079, A045778, A048993, A070826, A346426.
%K A346520 nonn,tabl
%O A346520 0,4
%A A346520 _Alois P. Heinz_, Jul 21 2021