A300847 a(n) = 12*binomial(n, 5).

0, 0, 0, 0, 0, 12, 72, 252, 672, 1512, 3024, 5544, 9504, 15444, 24024, 36036, 52416, 74256, 102816, 139536, 186048, 244188, 316008, 403788, 510048, 637560, 789360, 968760, 1179360, 1425060, 1710072, 2038932, 2416512, 2848032, 3339072, 3895584, 4523904, 5230764

Offset: 0

Author

Eric W. Weisstein, Mar 13 2018

Keywords

• nonn
• easy

Comments

Also the number of 5-cycles in the complete graph K_n for n >= 1.

Links

• F. Harary, B. Manvel, On the number of cycles in a graph, Matemat. casop. 21 (1971) 55-63, Theorem 2 for 5-cycles in complete graph.
• Eric Weisstein's World of Mathematics, Complete Graph
• Eric Weisstein's World of Mathematics, Graph Cycle
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000389, A052787.

Programs

Mathematica

Table[12 Binomial[n, 5], {n, 0, 20}]
12 Binomial[Range[0, 20], 5]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 0, 0, 12, 72}, {0, 20}]
CoefficientList[Series[12 x^5/(x - 1)^6, {x, 0, 20}], x]

PARI

a(n) = 12*binomial(n, 5); \\ Altug Alkan, Mar 13 2018

Formula

G.f.: 12*x^5/(x - 1)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) + 6*a(n-4) - a(n-5).

a(n) = A052787(n)/10 = 12*A000389(n).

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