A334356 -id:A334356 - cp's OEIS frontend

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

Andrew Howroyd, Apr 24 2020

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.

			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.

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

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

A334357 Number of nonequivalent proper colorings of the vertices of a 4D hypercube using at most n colors up to rotations and reflections of the cube.

Original entry on oeis.org

0, 1, 72, 7173, 610160, 28654530, 723903411, 11151501102, 117740542158, 928786063095, 5822688352360, 30338870238171, 135818642249082, 535712216425568, 1898338161488055, 6136965479845740, 18323823959847156, 51039512178104637, 133722394132080528

Offset: 1

Andrew Howroyd, Apr 24 2020

Adjacent vertices may not have the same color.

a(n) is the number of nonequivalent n-colorings of the tesseract graph up to graph isomorphism.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hypercube Graph
Eric Weisstein's World of Mathematics, Tesseract Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A128767, A158348, A334356, A334358.

a(n) = n*(n - 1)*(n^14 - 31*n^13 + 465*n^12 - 4471*n^11 + 30805*n^10 - 161035*n^9 + 659293*n^8 - 2149343*n^7 + 5610000*n^6 - 11666144*n^5 + 19009100*n^4 - 23485632*n^3 + 20729104*n^2 - 11646800*n + 3125472)/384.

a(n) = Sum_{k=1..16} n^k * A334358(4,16-k) / 384.

cp's OEIS Frontend

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.

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