A337406 Number of chiral pairs of colorings of the edges of a cube (or regular octahedron) using n or fewer colors.

0, 74, 10704, 345640, 5062600, 45246810, 288005144, 1430618784, 5881281480, 20827126650, 65370603320, 185725346664, 485325996064, 1181031257770, 2702889008400, 5863794289280, 12137528310384, 24099966466794

Offset

1,2

Comments

Each member of a chiral pair is a reflection, but not a rotation, of the other. Both the cube and the octahedron have 12 edges.

Links

Table of n, a(n) for n=1..18.

Formula

a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 2n^4 + 8) / 48.

a(n) = 74*C(n,2) + 10482*C(n,3) + 303268*C(n,4) + 3440700*C(n,5) + 19842840*C(n,6) + 65867760*C(n,7) + 133580160*C(n,8) + 168399000*C(n,9) + 128898000*C(n,10) + 54885600*C(n,11) + 9979200*C(n,12), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = (A060530(n) - A331351(n)) / 2 = A060530(n) - A199406(n) = A199406(n) - A331351(n).

G.f.: 2 * (37*x^2 + 4871*x^3 + 106130*x^4 + 691514*x^5 + 1692248*x^6 + 1692248*x^7 + 691514*x^8 + 106130*x^9 + 4871*x^10 + 37*x^11) / (1-x)^13.

Mathematica

Table[(n-1)n^2(n+1)(n^8+n^6-2n^4+8)/48, {n, 20}]

Crossrefs

Cf. A060530 (oriented), A199406 (unoriented), A331351 (achiral).

Row 3 of A337409 (orthotope edge colorings) and A337413 (orthoplex edge colorings).

Sequence in context: A262764 A107678 A093274 * A250906 A268090 A278378

Adjacent sequences: A337403 A337404 A337405 * A337407 A337408 A337409

Keyword

nonn,easy

Author

Robert A. Russell, Aug 26 2020

Status

approved