A290939 Number of 5-cycles in the n-triangular graph.

0, 0, 24, 312, 1584, 5376, 14448, 33264, 68544, 129888, 230472, 387816, 624624, 969696, 1458912, 2136288, 3055104, 4279104, 5883768, 7957656, 10603824, 13941312, 18106704, 23255760, 29565120, 37234080, 46486440, 57572424, 70770672, 86390304, 104773056, 126295488

Offset: 2

Author

Eric W. Weisstein, Aug 14 2017

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Johnson Graph
• Eric Weisstein's World of Mathematics, Triangular Graph
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A002417 (number of 3-cycles in the triangular graph), A151974 (4-cycles), A290940 (6-cycles).

Programs

Mathematica

Table[12/5 Binomial[n, 4] (n^2 + 7 n - 34), {n, 2, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 24, 312, 1584, 5376, 14448}, 20]
CoefficientList[Series[(24 (-x^2 - 6 x^3 + 4 x^4))/(-1 + x)^7, {x, 0, 20}], x]

PARI

a(n)=12*binomial(n, 4)*(n^2+7*n-34)/5 \\ Charles R Greathouse IV, Aug 14 2017

Formula

a(n) = 12/5 * binomial(n, 4) * (n^2 + 7*n - 34).

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

G.f.: (24 x^2 (-x^2 - 6 x^3 + 4 x^4))/(-1 + x)^7.