A181430 Number of distinct 5-card poker hands using n ranks and 4 unlabeled suits.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Adjacent sequences: A181427 A181428 A181429 * A181431 A181432 A181433

Keyword

nonn

Author

David Scambler, Oct 19 2010

Status

approved