A331359 Number of unoriented colorings of the edges of a tesseract with n available colors.

1, 11251322, 4825746875682, 48038446526132256, 60632984344185045000, 20725680132763499134746, 2876113738439693827763387, 206323339930086669420462592, 8941884949194537156253481511

Offset: 1

Author

Robert A. Russell, Jan 14 2020

Keywords

• nonn
• easy

Comments

A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. Two unoriented colorings are the same if congruent; chiral pairs are counted as one. Also the number of unoriented colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.

Links

• Index entries for linear recurrences with constant coefficients, signature (33, -528, 5456, -40920, 237336, -1107568, 4272048, -13884156, 38567100, -92561040, 193536720, -354817320, 573166440, -818809200, 1037158320, -1166803110, 1166803110, -1037158320, 818809200, -573166440, 354817320, -193536720, 92561040, -38567100, 13884156, -4272048, 1107568, -237336, 40920, -5456, 528, -33, 1)

Crossrefs

Cf. A331358 (oriented), A331360 (chiral), A331361 (achiral).

Cf. A063843 (simplex), A331355 (orthoplex), A338953 (24-cell), A338965 (120-cell, 600-cell).

Programs

Mathematica

Table[(48n^4 + 64n^6 + 164n^8 + 32n^12 + 35n^16 + 24n^18 + 16n^20 + n^32)/384, {n, 1, 25}]

Formula

a(n) = (48*n^4 + 64*n^6 + 164*n^8 + 32*n^12 + 35*n^16 + 24*n^18 + 16*n^20 + n^32) / 384.

a(n) = C(n,1) + 11251320*C(n,2) + 4825713121719*C(n,3) + 48019143606137456*C(n,4) + 60392840368910627325*C(n,5) + 20362602706881512104770*C(n,6) + 2732305589004849709507320*C(n,7) + 183891356981584237730865120*C(n,8) + 7186781660980022442696996900*C(n,9) + 179941570950595830458653229400*C(n,10) + 3092495918800698593432175049200*C(n,11) + 38355721930679608007610435655200*C(n,12) + 356388702642082232961224416430400*C(n,13) + 2552262270629849366778056301033600*C(n,14) + 14398742619650679721666540905600000*C(n,15) + 65081946248235516086688061276416000*C(n,16) + 238774230958640327164289928460608000*C(n,17) + 718111905257279424242461614311808000*C(n,18) + 1783226074397879202567353905547520000*C(n,19) + 3674025240535453233878734112386560000*C(n,20) + 6297428247692138525542940292326400000*C(n,21) + 8984640042458034573900227275929600000*C(n,22) + 10651431202956156039912718487654400000*C(n,23) + 10448264801973961157855568414105600000*C(n,24) + 8418935641672774875938561280000000000*C(n,25) + 5510766716064148076659382317056000000*C(n,26) + 2882400456553496466714071801856000000*C(n,27) + 1175640370514915165746352603136000000*C(n,28) + 360177463966855890088916582400000000*C(n,29) + 77945658076061560043023564800000000*C(n,30) + 10621166594979816972895518720000000*C(n,31) + 685236554514826901477130240000000*C(n,32), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = A331358(n) - A331360(n) = (A331358(n) - A331361(n)) / 2 = A331360(n) + A331361(n).