A063843 Number of n-multigraphs on 5 nodes.

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

Author

N. J. A. Sloane, Aug 25 2001

Keywords

• nonn
• easy

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

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
• Marko Riedel, Vertex colorings of the Petersen graph, Math StackExchange
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

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