OFFSET
1,3
COMMENTS
T(m,n) is the number of ways to distribute n*m different toys among m different kids so that each kid gets exactly n toys. For example, with n=3 and m=2, the 6 different toys, t1, t2, t3, t4, t5 and t6, can be distributed in exactly 20 ways among the 2 kids, k1 and k2, since there are C(6,3)=20 ways to choose the three toys for k1, with the other three toys going to k2. The proof for the general case is based on the identity C(n*m,n)*C(n*m-n,n)*...*C(n*m-n*(m-1),n) = (n*m)!/(n!)^m. - Dennis P. Walsh, Apr 12 2018
LINKS
Seiichi Manyama, Rows n = 1..26, flattened
Eric Weisstein's World of Mathematics, Macdonald's Constant-Term Conjecture
FORMULA
T(n, k) = (k*n)!/(n!)^k.
EXAMPLE
Triangle begins:
1;
1, 6;
1, 20, 1680;
1, 70, 34650, 63063000;
1, 252, 756756, 11732745024, 623360743125120;
MAPLE
T:= (m, n)-> (n*m)!/(m!)^n:
seq(seq(T(m, n), n=1..m), m=1..7); # Alois P. Heinz, Apr 12 2018
MATHEMATICA
Table[(n*k)!/(n!)^k, {n, 10}, {k, n}]//Flatten
PROG
(Magma) [Factorial(n*k)/(Factorial(n))^k: k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 26 2022
(SageMath)
def A120666(n, k): return gamma(n*k+1)/(factorial(n))^k
flatten([[A120666(n, k) for k in range(1, n+1)] for n in range(1, 11)]) # G. C. Greubel, Dec 26 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Aug 11 2006
EXTENSIONS
Edited by N. J. A. Sloane, Jun 17 2007
Offset corrected by Alois P. Heinz, Apr 12 2018
New name using formula by Joerg Arndt, Apr 15 2018
STATUS
approved