A291911 Number of 5-cycles in the n X n rook complement graph.

0, 0, 36, 3456, 77040, 800640, 5265540, 25514496, 99320256, 327836160, 951285060, 2488844160, 5980596336, 13384215936, 28197301860, 56398325760, 107825391360, 198142765056, 351580800996, 604675808640, 1011282960240, 1649187872640, 2628701385156

Offset: 1

Author

Eric W. Weisstein, Sep 05 2017

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Rook Complement Graph
• Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Crossrefs

Cf. A179058 (3-cycles), A291910 (4-cycles), A291912 (6-cycles).

Programs

Mathematica

Table[(n - 2)^2 (n - 1)^2 n^2 (4 + 2 n + 3 n^2 - 4 n^3 + n^4)/10, {n, 20}]
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 0, 36, 3456, 77040, 800640, 5265540, 25514496, 99320256, 327836160, 951285060}, 20]
CoefficientList[Series[-((36 x^2 (1 + 85 x + 1139 x^2 + 3815 x^3 + 3815 x^4 + 1139 x^5 + 85 x^6 + x^7))/(-1 + x)^11), {x, 0, 20}], x]

Formula

a(n) = (n - 2)^2*(n - 1)^2*n^2*(4 + 2*n + 3*n^2 - 4*n^3 + n^4)/10.

a(n) = 11*a(n-1) - 55*a(n-2) + 165*(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11).

G.f.: -((36 x^3 (1 + 85 x + 1139 x^2 + 3815 x^3 + 3815 x^4 + 1139 x^5 + 85 x^6 + x^7))/(-1 + x)^11).