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A167986 Triangle T(n,k) = Number of k-cycles on the graph of an n-orthoplex. n>=2, k>=3.

%I A167986 #42 Feb 16 2025 08:33:11
%S A167986 0,1,8,15,24,16,32,102,288,640,960,744,80,370,1584,5920,18240,43080,
%T A167986 69120,56256,160,975,5664,30080,141120,564120,1835520,4542336,7580160,
%U A167986 6385920,280,2121,15624,108080,684480,3876600,19138560,79805376
%N A167986 Triangle T(n,k) = Number of k-cycles on the graph of an n-orthoplex. n>=2, k>=3.
%C A167986 Row n contains 2n-2 elements.
%C A167986 The n-orthoplex is the dual polytope of the n-cube.
%C A167986 The orthoplex is also known as the cross-polytope.
%C A167986 Also the triangle of coefficients of the cocktail party graph cycle polynomials ordered from smallest to largest exponent starting with x^3. - _Eric W. Weisstein_
%H A167986 Andrew Howroyd, <a href="/A167986/b167986.txt">Table of n, a(n) for n = 2..871</a>
%H A167986 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrossPolytope.html">Cross Polytope</a>
%H A167986 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H A167986 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>
%F A167986 T(n,k) = Sum_{j=0..floor(k/2)} (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!/2. - _Andrew Howroyd_, May 09 2017
%e A167986 T(3,3) = 8, because in dimension n=3, the cross-polytope is the octahedron, which has 8 3-cycles in its graph.
%e A167986 Triangle starts
%e A167986    0,   1;
%e A167986    8,  15,   24,   16;
%e A167986   32, 102,  288,  640,   960,   744;
%e A167986   80, 370, 1584, 5920, 18240, 43080, 69120, 56256;
%e A167986   ...
%e A167986 In terms of cycle polynomials:
%e A167986    0*x^3 +    1*x^4;
%e A167986    8*x^3 +  15*x^4 +  24*x^5 +  16*x^6;
%e A167986   32*x^3 + 102*x^4 + 288*x^5 + 640*x^6 + 960*x^7 + 744*x^8;
%e A167986   ...
%t A167986 T[n_, k_]:= Sum[(-1)^j*Binomial[n, j]*Binomial[2*(n-j), k-2*j]*2^j*(k - j - 1)!/2, {j, 0, Floor[k/2]}];
%t A167986 Table[T[n, k], {n,2,7}, {k,3,2*n}]//Flatten (* _Jean-François Alcover_, Oct 08 2017, after _Andrew Howroyd_ *)
%t A167986 Table[Binomial[2n, k]*Gamma[k]*HypergeometricPFQ[{(1-k)/2, -k/2}, {1 - k, 1/2 -n}, -2]/2, {n,7}, {k,3,2n}]//Flatten (* _Eric W. Weisstein_, Mar 25 2020 *)
%o A167986 (PARI)
%o A167986 a(n,k)=sum(j=0,k\2, (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!)/2;
%o A167986 for (n=2,6,for (k=3,2*n, print1(a(n,k), ","));print); \\ _Andrew Howroyd_, May 09 2017
%o A167986 (Magma)
%o A167986 b:= func< n,k,j | (-1)^j*Binomial(n,j)*Binomial(2*(n-j),k-2*j)*2^(j-1)*Factorial(k-j-1) >;
%o A167986 A167986:= func< n,k | (&+[b(n,k,j): j in [0..Floor(k/2)]]) >;
%o A167986 [A167986(n,k): k in [3..2*n], n in [2..10]]; // _G. C. Greubel_, Jan 17 2023
%o A167986 (SageMath)
%o A167986 def A167986(n,k): return simplify(binomial(2*n, k)*gamma(k)*hypergeometric([(1-k)/2, -k/2], [1-k, 1/2-n], -2)/2)
%o A167986 flatten([[A167986(n,k) for k in range(3,2*n+1)] for n in range(2,11)]) # _G. C. Greubel_, Jan 17 2023
%Y A167986 Cf. A167987 (row sums).
%Y A167986 Cf. A085452 (2k-cycles on graph of n-cube).
%Y A167986 Cf. A144151 (k-cycles on (n-1)-simplex for k>3).
%K A167986 nonn,tabf
%O A167986 2,3
%A A167986 _Andrew Weimholt_, Nov 16 2009