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
Keywords
Links
- Eric Weisstein's World of Mathematics, Chordless Cycle
- Eric Weisstein's World of Mathematics, Johnson Graph
- Eric Weisstein's World of Mathematics, Triangular Graph
Crossrefs
Cf. A297670.
Programs
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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 *) -
PARI
a(n)=sum(k=2, n\2, binomial(n, 2*k+1)*(2*k)!/2) \\ Andrew Howroyd, Apr 28 2018
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
Extensions
a(9)-a(24) from Andrew Howroyd, Apr 28 2018
Comments