OFFSET
0,13
COMMENTS
Also number of ways to partition the multiset consisting of n copies each of 1, 2, ..., k into n multisets of size k.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals n=0..26)
EXAMPLE
A(4,2) = 3: (2*3)^4 = 1296 = 6*6*6*6 = 9*6*6*4 = 9*9*4*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) = 10: (2*3*5*7)^2 = 44100 = 210*210 = 225*196 = 294*150 = 315*140 = 350*126 = 441*100 = 490*90 = 525*84 = 735*60 = 1225*36.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 4, 10, 26, ...
1, 1, 2, 10, 70, 566, ...
1, 1, 3, 25, 465, 11131, ...
1, 1, 3, 49, 2505, 190131, ...
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) minus {1}))
end:
A:= (n, k)-> b(mul(ithprime(i), i=1..k)^n$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, Sum[If[d > i || PrimeOmega[d] != k, 0, b[n/d, d, k]], {d, Divisors[n] // Rest}]]; A[n_, k_] := Module[ {p = Product[Prime[i], {i, 1, k}]^n}, b[p, p, k]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
CROSSREFS
Main diagonal gives A334286.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 24 2015
STATUS
approved