A331355 - OEIS

%I #16 Mar 09 2024 12:02:25

%S 1,49127,740360358,733776248840,155261523065875,12340612271439081,

%T 498926608780739307,12298018390569089088,207726683413584244680,

%U 2604177120221402303875,25650403577338260144611,207023317470352041578712

%N Number of unoriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

%C A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. Two unoriented colorings are the same if congruent; chiral pairs are counted as one. Also the number of unoriented colorings of the square faces of a tesseract {4,3,3} with n available colors.

%H <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

%F a(n) = (48*n^3 + 64*n^4 + 44*n^6 + 84*n^7 + 56*n^8 + 12*n^9 + 5*n^12 +

%F    36*n^13 + 18*n^14 + 12*n^15 + 4*n^18 + n^24) / 384.

%F a(n) = C(n,1) + 49125*C(n, 2) + 740212980*C(n, 3) + 730815102166*C(n, 4) + 151600044933990*C(n, 5) + 11420034970306170*C(n, 6) + 415777158607920585*C(n, 7) + 8643499341510394200*C(n, 8) + 113988734942055623055*C(n, 9) + 1023002477284840979850*C(n, 10) + 6559265715033958749900*C(n, 11) + 31097943476763200314200*C(n, 12) + 111710751446923209781200*C(n, 13) + 309231173588248964052000*C(n, 14) + 666846649590586048584000*C(n, 15) + 1126625898539640346848000*C(n, 16) + 1492173541849975272288000*C(n, 17) + 1541987122059614438208000*C(n, 18) + 1229356526029003532160000*C(n, 19) + 741102367008078915840000*C(n, 20) + 326583680209195368960000*C(n, 21) + 99234043419574103040000*C(n, 22) + 18581137031073576960000*C(n, 23) + 1615751046180311040000*C(n, 24), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

%F a(n) = A331354(n) - A331356(n) = (A331354(n) + A331357(n)) / 2 = A331356(n) + A331357(n).

%t Table[(48 n^3 + 64 n^4 + 44 n^6 + 84 n^7 + 56 n^8 + 12 n^9 + 5 n^12 +

%t     36 n^13 + 18 n^14 + 12 n^15 + 4 n^18 + n^24)/384, {n, 1, 25}]

%Y Cf. A331354 (oriented), A331356 (chiral), A331357 (achiral).

%Y Other polychora: A063843 (5-cell), A331359 (8-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).

%Y Row 4 of A337412 (orthoplex edges, orthotope ridges) and A337888 (orthotope faces, orthoplex peaks).

%K nonn,easy

%O 1,2

%A _Robert A. Russell_, Jan 14 2020