OFFSET
0,13
COMMENTS
Also number of ways to partition the multiset consisting of k copies each of 1, 2, ..., n into n multisets of size k.
LINKS
Andrew Howroyd, Antidiagonals n = 0..27, flattened (antidiagonals 0..12 from Alois P. Heinz)
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. The array is on page 40.
Math StackExchange, Number of ways to partition 40 balls with 4 colors into 4 baskets
Marko Riedel, Maple program to compute array from cycle indices
EXAMPLE
A(4,2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*4.
A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8.
A(2,4) = 3: (2*3)^4 = 1296 = 36*36 = 54*24 = 81*16.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 2, 3, 3, 4, ...
1, 1, 5, 10, 23, 40, 73, ...
1, 1, 17, 93, 465, 1746, 5741, ...
1, 1, 73, 1417, 19834, 190131, 1398547, ...
1, 1, 388, 32152, 1532489, 43816115, 848597563, ...
MAPLE
with(numtheory):
b:= proc(n, i, k) option remember; `if`(n=1, 1,
add(`if`(d>i or bigomega(d)<>k, 0,
b(n/d, d, k)), d=divisors(n)))
end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n==1, 1, DivisorSum[n, If[#>i || PrimeOmega[#] != k, 0, b[n/#, #, k]]&]];
A[n_, k_] := b[p = Product[Prime[i], {i, 1, n}]^k, p, k];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *)
CROSSREFS
Main diagonal gives A334286.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 24 2015
STATUS
approved