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%I #24 Aug 19 2021 05:01:56
%S 1,1,1,2,1,2,5,3,3,5,15,9,5,9,15,52,31,18,18,31,52,203,120,70,40,70,
%T 120,203,877,514,299,172,172,299,514,877,4140,2407,1393,801,457,801,
%U 1393,2407,4140,21147,12205,7023,4025,2295,2295,4025,7023,12205,21147
%N Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k} into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i) into distinct factors; A(2,2) = 5: 2*3*6, 4*9, 3*12, 2*18, 36.
%H Alois P. Heinz, <a href="/A346517/b346517.txt">Antidiagonals n = 0..140, flattened</a>
%F A(n,k) = A045778(A002110(n)*A002110(k)).
%F A(n,k) = A(k,n).
%F A(n,k) = A322770(abs(n-k),min(n,k)).
%e A(2,2) = 5: 1122, 11|22, 1|122, 112|2, 1|12|2.
%e Square array A(n,k) begins:
%e 1, 1, 2, 5, 15, 52, 203, 877, ...
%e 1, 1, 3, 9, 31, 120, 514, 2407, ...
%e 2, 3, 5, 18, 70, 299, 1393, 7023, ...
%e 5, 9, 18, 40, 172, 801, 4025, 21709, ...
%e 15, 31, 70, 172, 457, 2295, 12347, 70843, ...
%e 52, 120, 299, 801, 2295, 6995, 40043, 243235, ...
%e 203, 514, 1393, 4025, 12347, 40043, 136771, 875936, ...
%e 877, 2407, 7023, 21709, 70843, 243235, 875936, 3299218, ...
%e ...
%p g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
%p `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
%p g(n/d, d-1)), d=divisors(n) minus {1, n}))
%p end:
%p p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
%p A:= (n, k)-> g(p(n)*p(k)$2):
%p seq(seq(A(n, d-n), n=0..d), d=0..10);
%p # second Maple program:
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p end:
%p A:= proc(n, k) option remember; `if`(n<k, A(k, n),
%p `if`(k=0, b(n), (A(n+1, k-1)-add(A(n-k+j, j)
%p *binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..10);
%t (* Q is A322770 *)
%t Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
%t A[n_, k_] := Q[Abs[n-k], Min[n, k]];
%t Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Aug 19 2021 *)
%Y Columns (or rows) k=0-10 give: A000110, A087648, A322773, A322774, A346897, A346898, A346899, A346900, A346901, A346902, A346903.
%Y Main diagonal gives A094574.
%Y First upper (or lower) diagonal gives A322771.
%Y Second upper (or lower) diagonal gives A322772.
%Y Antidiagonal sums give A346518.
%Y Cf. A002110, A045778, A322770, A346500.
%K nonn,tabl
%O 0,4
%A _Alois P. Heinz_, Jul 21 2021
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