A120666 Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.

1, 1, 6, 1, 20, 1680, 1, 70, 34650, 63063000, 1, 252, 756756, 11732745024, 623360743125120, 1, 924, 17153136, 2308743493056, 1370874167589326400, 2670177736637149247308800, 1, 3432, 399072960, 472518347558400, 3177459078523411968000, 85722533226982363751829504000, 7363615666157189603982585462030336000

Offset: 1

Author

Roger L. Bagula, Aug 11 2006

Keywords

• nonn
• tabl

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

Examples

			Triangle begins:
  1;
  1,   6;
  1,  20,   1680;
  1,  70,  34650,    63063000;
  1, 252, 756756, 11732745024, 623360743125120;
		

Links

• Seiichi Manyama, Rows n = 1..26, flattened
• Eric Weisstein's World of Mathematics, Macdonald's Constant-Term Conjecture

Crossrefs

Cf. A000984, A006480, A034841, A089759, A187783.

Programs

Magma

[Factorial(n*k)/(Factorial(n))^k: k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 26 2022

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

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

Formula

T(n, k) = (k*n)!/(n!)^k.

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