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

1,3

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

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

Sequence in context: A146383 A264313 A096130 * A050300 A185678 A286893

Adjacent sequences: A120663 A120664 A120665 * A120667 A120668 A120669

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