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A343658
Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.
14
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
OFFSET
1,5
COMMENTS
First differs from A343656 at A(4,2) = 6, A343656(4,2) = 5.
As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
FORMULA
A(n,k) = ((A000005(n), k)) = A007318(A000005(n) + k - 1, k).
T(n,k) = ((A000005(k), n - k)) = A007318(A000005(k) + n - k - 1, n - k).
EXAMPLE
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.
MATHEMATICA
multchoo[n_, k_]:=Binomial[n+k-1, k];
Table[multchoo[DivisorSigma[0, k], n-k], {n, 10}, {k, n}]
PROG
(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
CROSSREFS
Row k = 1 of the array is A000005.
Column n = 4 of the array is A000217.
Column n = 6 of the array is A000292.
Row k = 2 of the array is A184389.
The distinct products of these multisets are counted by A343656.
Antidiagonal sums of the array (or row sums of the triangle) are A343661.
A000312 = n^n.
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}.
Sequence in context: A343656 A278427 A077592 * A194005 A055794 A092905
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 29 2021
STATUS
approved