A331352 Number of chiral pairs of colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

0, 6, 405, 7904, 76880, 486522, 2300305, 8806336, 28725192, 82626270, 214744629, 513368064, 1144198952, 2402617490, 4792612545, 9142333696, 16768783408, 29707141878, 51023629173, 85234690080, 138859666848, 221139033962, 344955737489, 528003239616, 794250547000

Offset

1,2

Comments

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. The chiral colorings of its edges come in pairs, each the reflection of the other.

Links

Table of n, a(n) for n=1..25.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

a(n) = (24*n^2 - 50*n^3 + 20*n^4 + 15*n^6 - 10*n^7 + n^10) / 120.

a(n) = 6*C(n,2) + 387*C(n,3) + 6320*C(n,4) + 41350*C(n,5) + 135792*C(n,6) + 246540*C(n,7) + 252000*C(n,8) + 136080*C(n,9) + 30240*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = A331350(n) - A063843(n) = (A331350(n) - A331353(n)) / 2 = A063843(n) - A331353(n).

From Elmo R. Oliveira, Jun 02 2026: (Start)

G.f.: x^2*(6 + 339*x + 3779*x^2 + 11221*x^3 + 10717*x^4 + 3681*x^5 + 473*x^6 + 23*x^7 + x^8) / (1 - x)^11.

E.g.f.: exp(x)*x^2*(360 + 7740*x + 31600*x^2 + 41350*x^3 + 22632*x^4 + 5870*x^5 + 750*x^6 + 45*x^7 + x^8) / 120.

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). (End)

Mathematica

Table[(24n^2 - 50n^3 + 20n^4 + 15n^6 - 10n^7 + n^10)/120, {n, 1, 25}]

Crossrefs

Cf. A331350 (oriented), A063843 (unoriented), A331353 (achiral).

Other polychora: A331360 (8-cell), A331356 (16-cell), A338954 (24-cell), A338966 (120-cell, 600-cell).

Row 4 of A327085 (simplex edges and ridges) and A337885 (simplex faces and peaks).

Sequence in context: A029591 A151578 A172227 * A159015 A106206 A174773

Adjacent sequences: A331349 A331350 A331351 * A331353 A331354 A331355

Keyword

nonn,easy,changed

Author

Robert A. Russell, Jan 14 2020

Extensions

More terms from Elmo R. Oliveira, Jun 02 2026

Status

approved