A261174 Number of multigraphs on 4 unlabeled nodes with n edges where the edges can be of two colors.

1, 2, 9, 30, 90, 248, 650, 1560, 3560, 7680, 15786, 31076, 58905, 107768, 191180, 329664, 554038, 909558, 1461655, 2302950, 3563482, 5422392, 8124040, 11997648, 17482295, 25156872, 35779092, 50330364, 70072640, 96615760, 131999058, 178786960, 240186182, 320179470

Offset: 0

Author

Geoffrey Critzer, Aug 10 2015

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A050531 (case of 3 nodes).

Programs

Mathematica

Needs["Combinatorica`"];n = 4; nn = 25; CoefficientList[Series[PairGroupIndex[SymmetricGroup[n], s] /.Table[s[i] -> 1/(1 - x^i)^2, {i, 1, Binomial[n, 2]}], {x, 0, nn}], x]

PARI

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p, i->(1-x^i)^2)); s/n!}
{ Vec(G(4) + O(x^36)) } \\ Andrew Howroyd, Apr 18 2021

Formula

G.f.: (1 - 2*x + 5*x^2 + 2*x^3 + 10*x^4 + 12*x^5 + 32*x^6 + 20*x^7 + 56*x^8 + 20*x^9 + 32*x^10 + 12*x^11 + 10*x^12 + 2*x^13 + 5*x^14 - 2*x^15 + x^16)/((1 - x)^12*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^4). - Andrew Howroyd, Apr 18 2021

Extensions

Terms a(26) and beyond from Andrew Howroyd, Apr 18 2021