A244998 Number of ways for five teams of a World Cup football group to each have n goals for and n goals against.

1, 44, 870, 9480, 68290, 365936, 1573374, 5709120, 18107760, 51488800, 133748186, 321979164, 726436425, 1549758640, 3148837580, 6129352176, 11486265339, 20807609460, 36563769670, 62510325680, 104239453956, 169923049824, 271300238650, 424972948400

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Shalosh B. Ekhad, Doron Zeilberger, There are (1/30)(r+1)(r+2)(2r+3)(r^2+3r+5) Ways For the Four Teams of a World Cup Group to Each Have r Goals For and r Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1], arXiv:1407.1919, 2014.

Index entries for linear recurrences with constant coefficients, signature (12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1).

Formula

Shalosh B. Ekhad and Doron Zeilberger give an explicit formula for a(n).

G.f.: ( 1+32*x+408*x^2+1724*x^3+2765*x^4+1724*x^5+408*x^6+32*x^7+x^8 ) / (x-1)^12. - R. J. Mathar, Jan 31 2015

Maple

A244998 := proc(n)
    (n+1)*(n+2)*(n+3)/241920 ;
    %*(43*n^8 +688*n^7 +4934*n^6 +20680*n^5 +55907*n^4 +101272*n^3 +123436*n^2 +96240*n +40320) ;
end proc:

Mathematica

LinearRecurrence[{12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1}, {1, 44, 870, 9480, 68290, 365936, 1573374, 5709120, 18107760, 51488800, 133748186, 321979164}, 24] (* Jean-François Alcover, Dec 02 2017 *)

Crossrefs

Sequence in context: A297083 A282994 A295273 * A159948 A183756 A183750

Adjacent sequences: A244995 A244996 A244997 * A244999 A245000 A245001

Keyword

nonn

Author

N. J. A. Sloane, Jul 09 2014

Status

approved