|
%I #30 Mar 30 2020 08:42:00
%S 1,6,16,6,24,128,696,2112,5024,5376,1344,80,640,6720,68736,591200,
%T 4652160,32146800,185285120,865894848,3136412160,8315531200,
%U 14800412160,15448366080,7413471744,906545760,240,2560,39840,698112,12226560,203258880,3257746560
%N Triangle T(n,k) read by rows: T(n,k) = number of cycles of length 2k in the binary n-cube, for n >= 2, k = 2, 3, ..., 2^(n-1).
%C Row n contains 2^(n-1)-1 terms.
%C Also the triangle of even-order coefficients (odd coefficients are all 0) of the hypercube graph cycle polynomials ordered from smallest to largest exponent starting with x^4. - _Eric W. Weisstein_, Feb 05 2014
%D Initial terms computed by Daniele Degiorgi (danieled(AT)inf.ethz.ch).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>
%e Triangle begins:
%e 1,
%e 6, 16, 6,
%e 24, 128, 696, 2112, 5024, 5376, 1344,
%e 80, 640, 6720, 68736, 591200, 4652160, 32146800, 185285120, 865894848, 3136412160, 8315531200, 14800412160, 15448366080, 7413471744, 906545760,
%e ....
%e In terms of cycle polynomials:
%e x^4
%e 6*x^4 + 16*x^6 + 6*x^8
%e 24*x^4 + 128*x^6 + 696*x^8 + 2112*x^10 + 5024*x^12 + 5376*x^14 + 1344*x^16
%e ...
%t Table[Table[Length[FindCycle[HypercubeGraph[n], {k}, All]], {k, 4, 2^n, 2}], {n, 4}] // Flatten (* _Eric W. Weisstein_, Mar 23 2020 *)
%Y Cf. A066037, A001788. Row sums give A085408.
%K nonn,tabf,more,hard
%O 2,2
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 12 2003
%E Corrected by _Andrew Weimholt_, Nov 14 2009
%E Initial terms of T(6,k) from _Eric W. Weisstein_, Mar 23 2020
|
