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%I #16 Jan 11 2024 22:56:47
%S 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,
%T 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,
%U 20,6,4,1,1,11,10,45,8,84,6,35,10,10,2,1
%N Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.
%C First differs from A343656 at A(4,2) = 6, A343656(4,2) = 5.
%C As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.
%H Andrew Howroyd, <a href="/A343658/b343658.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%F A(n,k) = ((A000005(n), k)) = A007318(A000005(n) + k - 1, k).
%F T(n,k) = ((A000005(k), n - k)) = A007318(A000005(k) + n - k - 1, n - k).
%e Array begins:
%e k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
%e n=1: 1 1 1 1 1 1 1 1 1
%e n=2: 1 2 3 4 5 6 7 8 9
%e n=3: 1 2 3 4 5 6 7 8 9
%e n=4: 1 3 6 10 15 21 28 36 45
%e n=5: 1 2 3 4 5 6 7 8 9
%e n=6: 1 4 10 20 35 56 84 120 165
%e n=7: 1 2 3 4 5 6 7 8 9
%e n=8: 1 4 10 20 35 56 84 120 165
%e n=9: 1 3 6 10 15 21 28 36 45
%e Triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 3 2 1
%e 1 4 3 3 1
%e 1 5 4 6 2 1
%e 1 6 5 10 3 4 1
%e 1 7 6 15 4 10 2 1
%e 1 8 7 21 5 20 3 4 1
%e 1 9 8 28 6 35 4 10 3 1
%e 1 10 9 36 7 56 5 20 6 4 1
%e 1 11 10 45 8 84 6 35 10 10 2 1
%e For example, row n = 6 counts the following multisets:
%e {1,1,1,1,1} {1,1,1,1} {1,1,1} {1,1} {1} {}
%e {1,1,1,2} {1,1,3} {1,2} {5}
%e {1,1,2,2} {1,3,3} {1,4}
%e {1,2,2,2} {3,3,3} {2,2}
%e {2,2,2,2} {2,4}
%e {4,4}
%e 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.
%t multchoo[n_,k_]:=Binomial[n+k-1,k];
%t Table[multchoo[DivisorSigma[0,k],n-k],{n,10},{k,n}]
%o (PARI) A(n,k) = binomial(numdiv(n) + k - 1, k)
%o { for(n=1, 9, for(k=0, 8, print1(A(n,k), ", ")); print ) } \\ _Andrew Howroyd_, Jan 11 2024
%Y Row k = 1 of the array is A000005.
%Y Column n = 4 of the array is A000217.
%Y Column n = 6 of the array is A000292.
%Y Row k = 2 of the array is A184389.
%Y The distinct products of these multisets are counted by A343656.
%Y Antidiagonal sums of the array (or row sums of the triangle) are A343661.
%Y A000312 = n^n.
%Y A009998(n,k) = n^k (as an array, offset 1).
%Y A007318 counts k-sets of elements of {1..n}.
%Y A059481 counts k-multisets of elements of {1..n}.
%Y Cf. A000169, A062319, A067824, A143773, A146291, A176029, A285572, A326077, A327527, A334996, A343652, A343657.
%K nonn,tabl
%O 1,5
%A _Gus Wiseman_, Apr 29 2021
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