A334358 Irregular triangle read by rows: row n gives scaled coefficients of the chromatic polynomial corresponding to colorings of the n-hypercube graph up to automorphism, highest powers first, 0 <= k <= 2^n.

1, 0, 1, -1, 0, 1, -2, 3, -2, 0, 1, -12, 72, -256, 579, -812, 644, -216, 0, 1, -32, 496, -4936, 35276, -191840, 820328, -2808636, 7759343, -17276144, 30675244, -42494732, 44214736, -32375904, 14772272, -3125472, 0, 1, -80, 3160, -82080, 1575420, -23805776, 294640000

Offset: 0

Author

Andrew Howroyd, Apr 24 2020

Keywords

• sign
• tabf

Comments

The polynomials are scaled by a factor of n!*2^n to ensure integer coefficients. When evaluated at x = k, they give the number of non-isomorphic k-colorings of the n-hypercube graph under the automorphism group of the graph. The size of the automorphism group is n!*2^n. Colors may not be interchanged.

Examples

			Triangle begins:
  0 | 1, 0;
  1 | 1, -1, 0;
  2 | 1, -2, 3, -2, 0;
  3 | 1, -12, 72, -256, 579, -812, 644, -216, 0;
  ...
The corresponding polynomials are:
  x;
  (x^2 - x)/(1!*2^1);
  (x^4 - 2*x^3 + 3*x^2 - 2*x)/(2!*2^2);
  (x^8 - 12*x^7 + 72*x^6 - 256*x^5 + 579*x^4 - 812*x^3 + 644*x^2 - 216*x)/(3!*2^3);
  ...
The polynomial (x^4 - 2*x^3 + 3*x^2 - 2*x)/(2!*2^2) gives A002817 when evaluated at integer values of x.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..68 (rows 0..5)
• Eric Weisstein's World of Mathematics, Chromatic Polynomial
• Eric Weisstein's World of Mathematics, Hypercube Graph

Crossrefs

Cf. A002817, A334159, A334248, A334356, A334357.