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A343658
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Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.
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14
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1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 10, 2, 1, 1, 8, 7, 21, 5, 20, 3, 4, 1, 1, 9, 8, 28, 6, 35, 4, 10, 3, 1, 1, 10, 9, 36, 7, 56, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 84, 6, 35, 10, 10, 2, 1
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OFFSET
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1,5
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COMMENTS
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As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.
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LINKS
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FORMULA
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EXAMPLE
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Array begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
n=1: 1 1 1 1 1 1 1 1 1
n=2: 1 2 3 4 5 6 7 8 9
n=3: 1 2 3 4 5 6 7 8 9
n=4: 1 3 6 10 15 21 28 36 45
n=5: 1 2 3 4 5 6 7 8 9
n=6: 1 4 10 20 35 56 84 120 165
n=7: 1 2 3 4 5 6 7 8 9
n=8: 1 4 10 20 35 56 84 120 165
n=9: 1 3 6 10 15 21 28 36 45
Triangle begins:
1
1 1
1 2 1
1 3 2 1
1 4 3 3 1
1 5 4 6 2 1
1 6 5 10 3 4 1
1 7 6 15 4 10 2 1
1 8 7 21 5 20 3 4 1
1 9 8 28 6 35 4 10 3 1
1 10 9 36 7 56 5 20 6 4 1
1 11 10 45 8 84 6 35 10 10 2 1
For example, row n = 6 counts the following multisets:
{1,1,1,1,1} {1,1,1,1} {1,1,1} {1,1} {1} {}
{1,1,1,2} {1,1,3} {1,2} {5}
{1,1,2,2} {1,3,3} {1,4}
{1,2,2,2} {3,3,3} {2,2}
{2,2,2,2} {2,4}
{4,4}
Note that for n = 6, k = 4 in the triangle, the two multisets {1,4} and {2,2} represent the same divisor 4, so they are only counted once under A343656(4,2) = 5.
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MATHEMATICA
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multchoo[n_, k_]:=Binomial[n+k-1, k];
Table[multchoo[DivisorSigma[0, k], n-k], {n, 10}, {k, n}]
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PROG
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(PARI) A(n, k) = binomial(numdiv(n) + k - 1, k)
{ for(n=1, 9, for(k=0, 8, print1(A(n, k), ", ")); print ) } \\ Andrew Howroyd, Jan 11 2024
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CROSSREFS
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Column n = 4 of the array is A000217.
Column n = 6 of the array is A000292.
The distinct products of these multisets are counted by A343656.
Antidiagonal sums of the array (or row sums of the triangle) are A343661.
A009998(n,k) = n^k (as an array, offset 1).
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
Cf. A000169, A062319, A067824, A143773, A146291, A176029, A285572, A326077, A327527, A334996, A343652, A343657.
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KEYWORD
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AUTHOR
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STATUS
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approved
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