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 A346517 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. 15
 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, 120, 203, 877, 514, 299, 172, 172, 299, 514, 877, 4140, 2407, 1393, 801, 457, 801, 1393, 2407, 4140, 21147, 12205, 7023, 4025, 2295, 2295, 4025, 7023, 12205, 21147 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,k) = A045778(A002110(n)*A002110(k)). A(n,k) = A(k,n). A(n,k) = A322770(abs(n-k),min(n,k)). EXAMPLE A(2,2) = 5: 1122, 11|22, 1|122, 112|2, 1|12|2. Square array A(n,k) begins:     1,    1,    2,     5,    15,     52,    203,     877, ...     1,    1,    3,     9,    31,    120,    514,    2407, ...     2,    3,    5,    18,    70,    299,   1393,    7023, ...     5,    9,   18,    40,   172,    801,   4025,   21709, ...    15,   31,   70,   172,   457,   2295,  12347,   70843, ...    52,  120,  299,   801,  2295,   6995,  40043,  243235, ...   203,  514, 1393,  4025, 12347,  40043, 136771,  875936, ...   877, 2407, 7023, 21709, 70843, 243235, 875936, 3299218, ...   ... MAPLE g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+      `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,         g(n/d, d-1)), d=divisors(n) minus {1, n}))     end: p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end: A:= (n, k)-> g(p(n)*p(k)\$2): seq(seq(A(n, d-n), n=0..d), d=0..10); # second Maple program: b:= proc(n) option remember; `if`(n=0, 1,       add(b(n-j)*binomial(n-1, j-1), j=1..n))     end: A:= proc(n, k) option remember; `if`(n

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Last modified December 7 22:50 EST 2021. Contains 349590 sequences. (Running on oeis4.)