cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

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%I A303505 #19 Feb 16 2025 08:33:54
%S A303505 0,0,0,12,72,612,3552,34632,247824,3047544,26502624,396071604,
%T A303505 4055072664,71316639036,839706878016,16982482829136,225984627860256,
%U A303505 5165674068939696,76644407669629248,1953726395279874588,31974794507569558248,899186672783502993108,16089847137602083031328
%N A303505 Number of odd chordless cycles in the n-triangular (Johnson) graph.
%C A303505 Equivalently, the number of cycles in the complete graph with odd length greater than three. - _Andrew Howroyd_, Apr 28 2018
%H A303505 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>
%H A303505 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H A303505 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%F A303505 a(n) = Sum_{k=2, ceiling(n/2)-1} binomial(n, 2*k+1)*(2*k)!/2. - _Andrew Howroyd_, Apr 28 2018
%F A303505 a(n) ~ sqrt(Pi) * (exp(2) - (-1)^n) * n^(n - 1/2) / (2^(3/2) * exp(n+1)). - _Vaclav Kotesovec_, Apr 27 2024
%t A303505 Array[Sum[Binomial[#, 2 k + 1] (2 k)!/2, {k, 2, Ceiling[#/2] - 1}] &, 23, 2] (* _Michael De Vlieger_, Apr 28 2018 *)
%t A303505 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 A303505 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 A303505 (PARI) a(n)=sum(k=2, n\2, binomial(n, 2*k+1)*(2*k)!/2) \\ _Andrew Howroyd_, Apr 28 2018
%Y A303505 Cf. A297670.
%K A303505 nonn
%O A303505 2,4
%A A303505 _Eric W. Weisstein_, Apr 25 2018
%E A303505 a(9)-a(24) from _Andrew Howroyd_, Apr 28 2018