A085452 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).

1, 6, 16, 6, 24, 128, 696, 2112, 5024, 5376, 1344, 80, 640, 6720, 68736, 591200, 4652160, 32146800, 185285120, 865894848, 3136412160, 8315531200, 14800412160, 15448366080, 7413471744, 906545760, 240, 2560, 39840, 698112, 12226560, 203258880, 3257746560

Offset: 2

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 12 2003

Keywords

• nonn
• tabf
• more
• hard

Comments

Row n contains 2^(n-1)-1 terms.

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

Examples

			Triangle begins:
1,
6, 16, 6,
24, 128, 696, 2112, 5024, 5376, 1344,
80, 640, 6720, 68736, 591200, 4652160, 32146800, 185285120, 865894848, 3136412160, 8315531200, 14800412160, 15448366080, 7413471744, 906545760,
....
In terms of cycle polynomials:
x^4
6*x^4 + 16*x^6 + 6*x^8
24*x^4 + 128*x^6 + 696*x^8 + 2112*x^10 + 5024*x^12 + 5376*x^14 + 1344*x^16
...
		

References

• Initial terms computed by Daniele Degiorgi (danieled(AT)inf.ethz.ch).

Links

• Eric Weisstein's World of Mathematics, Cycle Polynomial
• Eric Weisstein's World of Mathematics, Hypercube Graph

Crossrefs

Cf. A066037, A001788. Row sums give A085408.

Programs

Mathematica

Table[Table[Length[FindCycle[HypercubeGraph[n], {k}, All]], {k, 4, 2^n, 2}], {n, 4}] // Flatten (* Eric W. Weisstein, Mar 23 2020 *)

Extensions

Corrected by Andrew Weimholt, Nov 14 2009

Initial terms of T(6,k) from Eric W. Weisstein, Mar 23 2020