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Number of distinct 5-card poker hands using n ranks and 4 unlabeled suits.
1

%I #11 Feb 25 2017 16:43:24

%S 0,0,6,57,272,901,2376,5362,10808,19998,34602,56727,88968,134459,

%T 196924,280728,390928,533324,714510,941925,1223904,1569729,1989680,

%U 2495086,3098376,3813130,4654130,5637411,6780312,8101527,9621156,11360756

%N Number of distinct 5-card poker hands using n ranks and 4 unlabeled suits.

%C Count of distinct hands, that is, with suit symmetries eliminated.

%H G. C. Greubel, <a href="/A181430/b181430.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = n * (51 * n^4 - 110 * n^3 + 165 * n^2 -130 * n + 24)/120 = n * (n-1) * (51*n^3 - 59*n^2 + 106*n - 24)/120.

%F General formula for n ranks, m unlabeled suits, and k-card hands is the coefficient of x^k in B_m(0!*(1+x)^n, 1!*(1+x^2)^n, ...,(m-1)!*(1+x^m)^n) / m!, where B_m() is the m-th complete Bell polynomial. It follows from the Redfield-Polya enumeration theorem. - _Max Alekseyev_ Oct 15 2010

%F G.f.: (6*x + 21*x^2 + 20*x^3 + 4*x^4)/(-1 + x)^6. - _Alexander R. Povolotsky_, Oct 21 2010

%t Join[{0}, CoefficientList[Series[(6*x + 21*x^2 + 20*x^3 + 4*x^4)/(-1 + x)^6, {x,0,50}], x]] (* _G. C. Greubel_, Feb 25 2017 *)

%o (PARI) x='x+O('x^50); concat([0,0], Vec((6*x + 21*x^2 + 20*x^3 + 4*x^4)/(-1 + x)^6)) \\ _G. C. Greubel_, Feb 25 2017

%K nonn

%O 0,3

%A _David Scambler_, Oct 19 2010