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A346517
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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.
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15
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..140, flattened
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FORMULA
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A(n,k) = A045778(A002110(n)*A002110(k)).
A(n,k) = A(k,n).
A(n,k) = A322770(abs(n-k),min(n,k)).
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EXAMPLE
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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, ...
...
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MAPLE
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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<k, A(k, n),
`if`(k=0, b(n), (A(n+1, k-1)-add(A(n-k+j, j)
*binomial(k-1, j), j=0..k-1)+A(n, k-1))/2))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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(* Q is A322770 *)
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}])];
A[n_, k_] := Q[Abs[n-k], Min[n, k]];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Aug 19 2021 *)
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CROSSREFS
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Columns (or rows) k=0-10 give: A000110, A087648, A322773, A322774, A346897, A346898, A346899, A346900, A346901, A346902, A346903.
Main diagonal gives A094574.
First upper (or lower) diagonal gives A322771.
Second upper (or lower) diagonal gives A322772.
Antidiagonal sums give A346518.
Cf. A002110, A045778, A322770, A346500.
Sequence in context: A025165 A345278 A212431 * A318354 A348373 A106480
Adjacent sequences: A346514 A346515 A346516 * A346518 A346519 A346520
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Jul 21 2021
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STATUS
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approved
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