OFFSET
1,5
COMMENTS
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
FORMULA
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}.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 29 2021
STATUS
approved