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A181430
Number of distinct 5-card poker hands using n ranks and 4 unlabeled suits.
1
0, 0, 6, 57, 272, 901, 2376, 5362, 10808, 19998, 34602, 56727, 88968, 134459, 196924, 280728, 390928, 533324, 714510, 941925, 1223904, 1569729, 1989680, 2495086, 3098376, 3813130, 4654130, 5637411, 6780312, 8101527, 9621156, 11360756
OFFSET
0,3
COMMENTS
Count of distinct hands, that is, with suit symmetries eliminated.
LINKS
FORMULA
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.
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
G.f.: (6*x + 21*x^2 + 20*x^3 + 4*x^4)/(-1 + x)^6. - Alexander R. Povolotsky, Oct 21 2010
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A053421 A083696 A288680 * A281557 A227813 A221575
KEYWORD
nonn
AUTHOR
David Scambler, Oct 19 2010
STATUS
approved