A342075 Number of n-colorings of the vertices of the 7-dimensional cross polytope such that no two adjacent vertices have the same color.

0, 0, 0, 0, 0, 0, 0, 5040, 322560, 10342080, 216518400, 3261535200, 37026823680, 325474269120, 2264594492160, 12789814237200, 60389186457600, 245221330273920, 877374833287680, 2821277454690240, 8284633867238400, 22503569636419200, 57135310310453760

Offset: 0

Author

Peter Kagey, Feb 27 2021

Keywords

• nonn

Links

• Peter Kagey, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Crossrefs

Analogous for k-dimensional cross polytope: A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342074 (k=6).

Cf. A334279, A342088.

Programs

Mathematica

p = ChromaticPolynomial[CompleteGraph[Table[2, 7]], x];
Table[p /. x -> n, {n, 0, 50}]

Formula

a(n) = -3597143040*n + 11590795728*n^2 - 15837356724*n^3 + 12355698460*n^4 - 6212542175*n^5 + 2144307578*n^6 - 526197678*n^7 + 93450369*n^8 - 12064836*n^9 + 1122618*n^10 - 73423*n^11 + 3206*n^12 - 84*n^13 + n^14.

a(n) = (n - 6)*(n - 5)*(n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(n^7 - 63 n^6 + 1708 n^5 - 25795 n^4 + 234094 n^3 - 1275281 n^2 + 3858049 n - 4996032).

a(n) = Sum_{i=1..14} A334279(7,i)*n^i.

From Chai Wah Wu, Jan 19 2024: (Start)

a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15) for n > 14.

G.f.: x^7*(-52370755920*x^7 - 27190754640*x^6 - 6557740560*x^5 - 959792400*x^4 - 92962800*x^3 - 6032880*x^2 - 246960*x - 5040)/(x - 1)^15. (End)