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

%I A120666 #43 Feb 16 2025 08:33:01
%S A120666 1,1,6,1,20,1680,1,70,34650,63063000,1,252,756756,11732745024,
%T A120666 623360743125120,1,924,17153136,2308743493056,1370874167589326400,
%U A120666 2670177736637149247308800,1,3432,399072960,472518347558400,3177459078523411968000,85722533226982363751829504000,7363615666157189603982585462030336000
%N A120666 Triangle read by rows:  T(n, k) = (n*k)!/(n!)^k.
%C A120666 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
%H A120666 Seiichi Manyama, <a href="/A120666/b120666.txt">Rows n = 1..26, flattened</a>
%H A120666 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MacdonaldsConstant-TermConjecture.html">Macdonald's Constant-Term Conjecture</a>
%F A120666 T(n, k) = (k*n)!/(n!)^k.
%e A120666 Triangle begins:
%e A120666   1;
%e A120666   1,   6;
%e A120666   1,  20,   1680;
%e A120666   1,  70,  34650,    63063000;
%e A120666   1, 252, 756756, 11732745024, 623360743125120;
%p A120666 T:= (m, n)-> (n*m)!/(m!)^n:
%p A120666 seq(seq(T(m, n), n=1..m), m=1..7);  # _Alois P. Heinz_, Apr 12 2018
%t A120666 Table[(n*k)!/(n!)^k, {n,10}, {k,n}]//Flatten
%o A120666 (Magma) [Factorial(n*k)/(Factorial(n))^k: k in [1..n], n in [1..10]]; // _G. C. Greubel_, Dec 26 2022
%o A120666 (SageMath)
%o A120666 def A120666(n,k): return gamma(n*k+1)/(factorial(n))^k
%o A120666 flatten([[A120666(n,k) for k in range(1,n+1)] for n in range(1,11)]) # _G. C. Greubel_, Dec 26 2022
%Y A120666 Cf. A000984, A006480, A034841, A089759, A187783.
%K A120666 nonn,tabl
%O A120666 1,3
%A A120666 _Roger L. Bagula_, Aug 11 2006
%E A120666 Edited by _N. J. A. Sloane_, Jun 17 2007
%E A120666 Offset corrected by _Alois P. Heinz_, Apr 12 2018
%E A120666 New name using formula by _Joerg Arndt_, Apr 15 2018