A007010 Number of 4-voter voting schemes with n linearly ranked choices.

1, 12, 81, 372, 1332, 3984, 10420, 24540, 53145, 107436, 205065, 372792, 649936, 1092672, 1779408, 2817288, 4350105, 6567660, 9716905, 14114892, 20163924, 28368912, 39357396, 53902212, 72947329, 97636812, 129347505, 169725360, 220726080, 284659968, 364241728

Offset: 1

Author

Daniel E. Loeb

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [broken link]
• Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1).

Programs

Mathematica

LinearRecurrence[{6,-12,2,27,-36,0,36,-27,-2,12,-6,1},{1,12,81,372,1332,3984,10420,24540,53145,107436,205065,372792},40] (* Harvey P. Dale, Feb 12 2023 *)

PARI

Vec(x*(1+6*x+21*x^2+28*x^3+21*x^4+6*x^5+x^6)/((1+x)^3*(1-x)^9) + O(x^100)) \\ Colin Barker, Jan 07 2016

Formula

G.f.: x*(1+6*x+21*x^2+28*x^3+21*x^4+6*x^5+x^6)/((1+x)^3*(1-x)^9). - Ralf Stephan, Apr 23 2004

From Colin Barker, Jan 07 2016: (Start)

a(n) = (n^8+16*n^7+106*n^6+376*n^5+784*n^4+1024*n^3+864*n^2+384*n)/3840 for n even.

a(n) = (n^8+16*n^7+106*n^6+376*n^5+784*n^4+1024*n^3+894*n^2+504*n+135)/3840 for n odd.

(End)