cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A063843 Number of n-multigraphs on 5 nodes.

Original entry on oeis.org

0, 1, 34, 792, 10688, 90005, 533358, 2437848, 9156288, 29522961, 84293770, 217993600, 519341472, 1154658869, 2420188694, 4821091920, 9187076352, 16837177281, 29809183410, 51172613512, 85448030080, 139159855989, 221554769150, 345523218536, 528767663040
Offset: 0

Views

Author

N. J. A. Sloane, Aug 25 2001

Keywords

Comments

Equivalently, number of ways to color edges of complete graph on 5 nodes with n colors, under action of symmetric group S_5, of order 120, with cycle index on edges given by (1/120)*(24*x5^2 + 30*x2*x4^2 + 20*x3^3*x1 + 20*x3*x6*x1 + 15*x1^2*x2^4 + 10*x1^4*x2^3 + x1^10). Setting all x_i = n gives the sequence.
Number of vertex colorings of the Petersen graph. Marko Riedel, Mar 24 2016
Number of unoriented colorings of the 10 triangular edges or triangular faces of a pentachoron, Schläfli symbol {3,3,3}, using n or fewer colors. Also called a 5-cell or 4-simplex. - Robert A. Russell, Oct 17 2020

Links

Crossrefs

Cf. A063842. A row of A063841.
Cf. A331350 (oriented), A331352 (chiral), A331353 (achiral), A000389(n+4) (vertices and facets)
Other polychora: A331359 (8-cell), A331355 (16-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A327084 (simplex edges and ridges) and A337884 (simplex faces and peaks).

Programs

  • Maple
    f:=n-> 1/120*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10);
  • Mathematica
    Table[(24n^2+50n^3+20n^4+15n^6+10n^7+n^10)/120,{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,1,34,792,10688,90005,533358,2437848,9156288,29522961,84293770},30] (* Harvey P. Dale, Oct 20 2012 *)
  • PARI
    a(n)=n^2*(n^8+10*n^5+15*n^4+20*n^2+50*n+24)/120 \\ Charles R Greathouse IV, Jan 20 2012

Formula

a(n) = (1/120)*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10).
a(n+1) = (1/5!)*(n^10 + 10*n^9 + 45*n^8 + 130*n^7 + 295*n^6 + 552*n^5 + 805*n^4 + 900*n^3 + 774*n^2 + 448*n + 120).
G.f. = (1 + 23*x + 473*x^2 + 3681*x^3 + 10717*x^4 + 11221*x^5 + 3779*x^6 + 339*x^7 + 6*x^8)/(1-x)^11. - M. F. Hasler, Jan 19 2012
a(0)=0, a(1)=1, a(2)=34, a(3)=792, a(4)=10688, a(5)=90005, a(6)=533358, a(7)=2437848, a(8)=9156288, a(9)=29522961, a(10)=84293770, 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). - Harvey P. Dale, Oct 20 2012
From Robert A. Russell, Oct 17 2020: (Start)
a(n) = A331350(n) - A331352(n) = (A331350(n) + A331353(n)) / 2 = A331352(n) + A331353(n).
a(n) = 1*C(n,1) + 32*C(n,2) + 693*C(n,3) + 7720*C(n,4) + 44150*C(n,5) + 138312*C(n,6) + 247380*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 unoriented colorings using exactly k colors. (End)

Extensions

More terms from Vladeta Jovovic, Sep 02 2001

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

Original entry on oeis.org

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

Views

Author

Robert A. Russell, Jan 14 2020

Keywords

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).

A331355 Number of unoriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

1, 49127, 740360358, 733776248840, 155261523065875, 12340612271439081, 498926608780739307, 12298018390569089088, 207726683413584244680, 2604177120221402303875, 25650403577338260144611, 207023317470352041578712
Offset: 1

Views

Author

Robert A. Russell, Jan 14 2020

Keywords

Comments

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.

Links

Crossrefs

Cf. A331354 (oriented), A331356 (chiral), A331357 (achiral).
Other polychora: A063843 (5-cell), A331359 (8-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A337412 (orthoplex edges, orthotope ridges) and A337888 (orthotope faces, orthoplex peaks).

Programs

  • Mathematica
    Table[(48 n^3 + 64 n^4 + 44 n^6 + 84 n^7 + 56 n^8 + 12 n^9 + 5 n^12 +
    36 n^13 + 18 n^14 + 12 n^15 + 4 n^18 + n^24)/384, {n, 1, 25}]

Formula

a(n) = (48*n^3 + 64*n^4 + 44*n^6 + 84*n^7 + 56*n^8 + 12*n^9 + 5*n^12 +
36*n^13 + 18*n^14 + 12*n^15 + 4*n^18 + n^24) / 384.
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.
a(n) = A331354(n) - A331356(n) = (A331354(n) + A331357(n)) / 2 = A331356(n) + A331357(n).

A338949 Number of unoriented colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 18736, 249563343, 245072692820, 51780391393325, 4114243321427946, 166320182540310771, 4099464588809407728, 69243270244113372390, 868065984969662449300, 8550173137863803682016, 69007957379144017626756
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual.

Links

Crossrefs

Cf. A338948 (oriented), A338950 (chiral), A338951 (achiral), A338953 (edges, faces), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338965 (120-cell, 600-cell).

Programs

  • Mathematica
    Table[(96n^2+144n^3+144n^4+140n^6+300n^7+120n^8+36n^9+45n^12+84n^13+18n^14+12n^15+12n^18+n^24)/1152,{n,15}]
LinearRecurrence[{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},{1,18736,249563343,245072692820,51780391393325,4114243321427946,166320182540310771,4099464588809407728,69243270244113372390,868065984969662449300,8550173137863803682016,69007957379144017626756,471182396311499869193288,2790108355121570273031710,14612960014479438426745050,68774495831757984888966336,294660451484256436406752191,1161683435155207577365494648,4252399462403852518286044405,14563558286595288907896687700,46968928774940328123724865031,143447144215320073513164583826,416884377543198363455158598933,1157756823443195554136397711600,3083952997773835021725260467500},20] (* Harvey P. Dale, Mar 24 2024 *)

Formula

a(n) = (96*n^2 + 144*n^3 + 144*n^4 + 140*n^6 + 300*n^7 + 120*n^8 + 36*n^9 + 45*n^12 + 84*n^13 + 18*n^14 + 12*n^15 + 12*n^18 + n^24) / 1152.
a(n) = 1*C(n,1) + 18734*C(n,2) + 249507138*C(n,3) + 244074551860*C(n,4) + 50557523375300*C(n,5) + 3807232072474470*C(n,6) + 138599298699649830*C(n,7) + 2881219380682352640*C(n,8) + 37996512548398853085*C(n,9) + 341001760994302265550*C(n,10) + 2186424231002014796100*C(n,11) + 10365985337974980021000*C(n,12) + 37236922591331944681200*C(n,13) + 103077062953464218018400*C(n,14) + 222282219864764987928000*C(n,15) + 375541967632270447008000*C(n,16) + 497391180994576316448000*C(n,17) + 513995707397665741248000*C(n,18) + 409785508676334510720000*C(n,19) + 247034122336026305280000*C(n,20) + 108861226736398456320000*C(n,21) + 33078014473191367680000*C(n,22) + 6193712343691192320000*C(n,23) + 538583682060103680000*C(n,24), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A338948(n) - A338950(n) = (A338948(n) + A338951(n)) / 2 = A338950(n) + A338951(n).

A338952 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 137548893254081168086800768, 11046328890861011039111168376671536861388643, 10897746068379654103881579020805286236644252743361724416
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the rotation group of the 24-cell. They divide into 20 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^96 6+6+36+36 x_4^24
72 x_1^4x_2^46 32 x_2^3x_6^15
1+18 x_2^48 8+8+32 x_6^16
32 x_1^6x_3^30 72+72 x_8^12
8+8+32 x_3^32 48+48 x_12^8

Links

Crossrefs

Cf. A338953 (unoriented), A338954 (chiral), A338955 (achiral), A338956 (exactly n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338964 (120-cell, 600-cell).

Programs

  • Mathematica
    Table[(96n^8+144n^12+48n^16+32n^18+84n^24+48n^32+32n^36+19n^48+72n^50+n^96)/576,{n,15}]

Formula

a(n) = (96*n^8 + 144*n^12 + 48*n^16 + 32*n^18 + 84*n^24 + 48*n^32 + 32*n^36 + 19*n^48 + 72*n^50 + n^96) / 576.
a(n) = Sum_{j=1..Min(n,96)} A338956(n) * binomial(n,j).
a(n) = A338953(n) + A338954(n) = 2*A338953(n) - A338955(n) = 2*A338954(n) + A338955(n).

A338954 Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

68774446614978208476646592, 5523164445430505077912054084256733211946217, 5448873034189827051926943172520863487560602391778344960, 10956401461402941741829554371669666304159415287557559324930859375
Offset: 2

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual.

Links

Crossrefs

Cf. A338952 (oriented), A338953 (unoriented), A338955 (achiral), A338958 (exactly n colors), A338950 (vertices, facets), A331352 (5-cell), A331360 (8-cell edges, 16-cell faces), A331356 (16-cell edges, 8-cell faces), A338966 (120-cell, 600-cell).

Programs

  • Mathematica
    Table[(96n^8+144n^12-48n^16-64n^18-192n^20-60n^24+48n^32+32n^36-5n^48+72n^50-12n^52-12n^60+n^96)/1152,{n,2,15}]

Formula

a(n) = (96*n^8 + 144*n^12 - 48*n^16 - 64*n^18 - 192*n^20 - 60*n^24 +
48*n^32 + 32*n^36 - 5*n^48 + 72*n^50 - 12*n^52 - 12*n^60 + n^96) / 1152.
a(n) = Sum_{j=2..Min(n,96)} A338958(n) * binomial(n,j).
a(n) = A338952(n) - A338953(n) = (A338952(n) - A338955(n)) / 2 = A338953(n) - A338955(n).

A338955 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 24124751133507584, 883287060208158070437496209, 27692675763559261523047959805034496, 18070082615414169898334284655914306640625, 1018202231744161700740376040914469837333037056
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Count Odd Cycle Indices Count Odd Cycle Indices
12 x_1^24x_2^36 96 x_1^2x_2^2x_3^2x_6^14
12 x_1^8x_2^44 96 x_3^8x_6^12
12+12 x_3^48 96 x_2^3x_6^15
72+72 x_4^24 96 x_6^16

Links

Crossrefs

Cf. A338952 (oriented), A338953 (unoriented), A338954 (chiral), A338959 (exactly n colors), A338951 (vertices, facets), A331353 (5-cell), A331361 (8-cell edges, 16-cell faces), A331357 (16-cell edges, 8-cell faces), A338967 (120-cell, 600-cell).

Programs

  • Mathematica
    Table[(8n^16+8n^18+16n^20+12n^24+2n^48+n^52+n^60)/48,{n,15}]

Formula

a(n) = (8*n^16 + 8*n^18 + 16*n^20 + 12*n^24 + 2*n^48 + n^52 + n^60) / 48.
a(n) = Sum_{j=1..Min(n,60)} A338959(n) * binomial(n,j).
a(n) = 2*A338953(n) - A338952(n) = A338952(n) - 2*A338954(n) = A338953(n) - A338954(n).

A338957 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

Original entry on oeis.org

1, 68774446639102959610154174, 5523164445430505754875774375105924818979901, 5448873034167734394172913824852272971748608894646534804, 10956401434158576570935668826433407535831446552957081921713485225
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

Links

Crossrefs

Cf. A338956 (oriented), A338958 (chiral), A338959 (achiral), A338953 (up to n colors), A338949 (vertices, facets), A063843 (5-cell), A331359 (8-cell edges, 16-cell faces), A331355 (16-cell edges, 8-cell faces), A338981 (120-cell, 600-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (* binomial series *)
Drop[CoefficientList[bp[8]/12+bp[12]/8+bp[16]/8+bp[18]/9+bp[20]/6+19bp[24]/96+bp[32]/24+bp[36]/36+43bp[48]/1152+bp[50]/16+bp[52]/96+bp[60]/96+bp[96]/1152,x],1]

Formula

A338953(n) = Sum_{j=1..Min(n,96)} a(n) * binomial(n,j).
a(n) = A338956(n) - A338958(n) = (A338956(n) + A338959(n)) / 2 = A338958(n) + A338959(n).
Showing 1-8 of 8 results.

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