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.
Array giving number of connected graphs on n nodes with chromatic number k begins: n=...1...2...3...4....5....6.....7......8........9........10 k.------------------------------------------------------------ 2|...0...1...1...3....5...17....44....182......730......4032 3|...0...0...1...2...12...64...475...5036....80947...2010328 4|...0...0...0...1....3...26...282...5009...149551...7694428 5|...0...0...0...0....1....4....46....809....27794...1890221 6|...0...0...0...0....0....1.....5.....74.....1940....113272 7|...0...0...0...0....0....0.....1......6......110......4125 8|...0...0...0...0....0....0.....0......1........7.......156
a(1) = 2 because the single node can either be in the distinguished bipartite block or not. a(2) = 1 because the only connected bipartite graph on two nodes is the complete graph on two nodes. a(3) = 2 because the only connected bipartite graph on three nodes is the path graph on three nodes and there is a choice about which nodes are in the distinguished block.
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
Join[{1}, EULERi[Array[b, 23]]] (* Jean-François Alcover, Sep 13 2018, after Alois P. Heinz in A049312 *)
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i-1}] + Total @ Quotient[v+1, 2]
b[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Join[{1}, EULERi[Array[b, 20]]] (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)
Table of number of connected triangle-free graphs on n nodes with chromatic number k begins: (The first row is the same as the number of connected bipartite graphs on n nodes, A005142) n=...1...2...3...4...5....6....7.....8.....9.....10......11.......12.........13 k.----------------------------------------------------------------------------- 2|...0...1...1...3...5...17...44...182...730...4032...25598...212780....2241730 (A005142) 3|...0...0...0...0...1....2...15....85...650...5800...65243...931258...17182237 (the present sequence) 4|...0...0...0...0...0....0....0.....0.....0......0.......1.......23.......1085 (A126741)
Comments