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%I #18 Apr 27 2024 05:51:47
%S 0,0,0,12,72,612,3552,34632,247824,3047544,26502624,396071604,
%T 4055072664,71316639036,839706878016,16982482829136,225984627860256,
%U 5165674068939696,76644407669629248,1953726395279874588,31974794507569558248,899186672783502993108,16089847137602083031328
%N Number of odd chordless cycles in the n-triangular (Johnson) graph.
%C Equivalently, the number of cycles in the complete graph with odd length greater than three. - _Andrew Howroyd_, Apr 28 2018
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%F a(n) = Sum_{k=2, ceiling(n/2)-1} binomial(n, 2*k+1)*(2*k)!/2. - _Andrew Howroyd_, Apr 28 2018
%F a(n) ~ sqrt(Pi) * (exp(2) - (-1)^n) * n^(n - 1/2) / (2^(3/2) * exp(n+1)). - _Vaclav Kotesovec_, Apr 27 2024
%t Array[Sum[Binomial[#, 2 k + 1] (2 k)!/2, {k, 2, Ceiling[#/2] - 1}] &, 23, 2] (* _Michael De Vlieger_, Apr 28 2018 *)
%t 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 *)
%t 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 *)
%o (PARI) a(n)=sum(k=2, n\2, binomial(n, 2*k+1)*(2*k)!/2) \\ _Andrew Howroyd_, Apr 28 2018
%Y Cf. A297670.
%K nonn
%O 2,4
%A _Eric W. Weisstein_, Apr 25 2018
%E a(9)-a(24) from _Andrew Howroyd_, Apr 28 2018
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