A303505 Number of odd chordless cycles in the n-triangular (Johnson) graph.

0, 0, 0, 12, 72, 612, 3552, 34632, 247824, 3047544, 26502624, 396071604, 4055072664, 71316639036, 839706878016, 16982482829136, 225984627860256, 5165674068939696, 76644407669629248, 1953726395279874588, 31974794507569558248, 899186672783502993108, 16089847137602083031328

Offset

2,4

Comments

Equivalently, the number of cycles in the complete graph with odd length greater than three. - Andrew Howroyd, Apr 28 2018

Links

Table of n, a(n) for n=2..24.

Eric Weisstein's World of Mathematics, Chordless Cycle

Eric Weisstein's World of Mathematics, Johnson Graph

Eric Weisstein's World of Mathematics, Triangular Graph

Formula

a(n) = Sum_{k=2, ceiling(n/2)-1} binomial(n, 2*k+1)*(2*k)!/2. - Andrew Howroyd, Apr 28 2018

a(n) ~ sqrt(Pi) * (exp(2) - (-1)^n) * n^(n - 1/2) / (2^(3/2) * exp(n+1)). - Vaclav Kotesovec, Apr 27 2024

Mathematica

Array[Sum[Binomial[#, 2 k + 1] (2 k)!/2, {k, 2, Ceiling[#/2] - 1}] &, 23, 2] (* Michael De Vlieger, Apr 28 2018 *)
Table[Sum[Binomial[n, 2 k + 1] (2 k)!/2, {k, 2, Ceiling[n/2] - 1}], {n, 2, 20}] (* Eric W. Weisstein, Apr 29 2018 *)
Join[{0, 0, 0}, Table[12 Binomial[n, 5] HypergeometricPFQ[{1, 5/2, (5 - n)/2, 3 - n/2}, {7/2}, 4], {n, 5, 20}]] (* Eric W. Weisstein, Apr 29 2018 *)

Prog

(PARI) a(n)=sum(k=2, n\2, binomial(n, 2*k+1)*(2*k)!/2) \\ Andrew Howroyd, Apr 28 2018

Crossrefs

Cf. A297670.

Sequence in context: A030235 A088166 A138402 * A320660 A367700 A108734

Adjacent sequences: A303502 A303503 A303504 * A303506 A303507 A303508

Keyword

nonn

Author

Eric W. Weisstein, Apr 25 2018

Extensions

a(9)-a(24) from Andrew Howroyd, Apr 28 2018

Status

approved