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A120666
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Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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T(n, k) = (k*n)!/(n!)^k.
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EXAMPLE
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Triangle begins:
1;
1, 6;
1, 20, 1680;
1, 70, 34650, 63063000;
1, 252, 756756, 11732745024, 623360743125120;
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MAPLE
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T:= (m, n)-> (n*m)!/(m!)^n:
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MATHEMATICA
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Table[(n*k)!/(n!)^k, {n, 10}, {k, n}]//Flatten
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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