%I #26 Aug 05 2021 16:12:58
%S 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 4,877,877,514,244,100,38,14,5,4140,4140,2407,1112,442,161,56,19,6,
%U 21147,21147,12205,5516,2134,750,249,80,25,8,115975,115975,66491,29505,11147,3799,1213,372,111,33,10
%N 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 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 Alois P. Heinz, <a href="/A346520/b346520.txt">Antidiagonals n = 0..140, flattened</a>
%F A(n,k) = A045778(A000079(n)*A070826(k+1)).
%F A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).
%e 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 Square array A(n,k) begins:
%e 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
%e 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
%e 1, 3, 9, 31, 120, 514, 2407, 12205, 66491, ...
%e 2, 5, 16, 59, 244, 1112, 5516, 29505, 168938, ...
%e 2, 7, 25, 100, 442, 2134, 11147, 62505, 373832, ...
%e 3, 10, 38, 161, 750, 3799, 20739, 121141, 752681, ...
%e 4, 14, 56, 249, 1213, 6404, 36332, 220000, 1413937, ...
%e 5, 19, 80, 372, 1887, 10340, 60727, 379831, 2516880, ...
%e 6, 25, 111, 539, 2840, 16108, 97666, 629346, 4288933, ...
%e ...
%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, k)-> add(S(k, j)*b(n, j), j=0..k):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%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_, k_] := Sum[S[k, j]*b[n, j], {j, 0, k}];
%t Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jul 31 2021, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000009, A036469, A346822, A346823, A346824, A346825, A346826, A346827, A346828, A346829, A346830.
%Y Rows n=0+1,2-10 give: A000110, A087648, A346813, A346814, A346815, A346816, A346817, A346818, A346819, A346820.
%Y Main diagonal gives A346519.
%Y Antidiagonal sums give A346521.
%Y Cf. A000040, A000079, A045778, A048993, A070826, A346426.
%K nonn,tabl
%O 0,4
%A _Alois P. Heinz_, Jul 21 2021