A001433 Number of graphs with n nodes and n-1 edges.

1, 1, 1, 3, 6, 15, 41, 115, 345, 1103, 3664, 12763, 46415, 175652, 691001, 2821116, 11932174, 52211412, 236007973, 1100528508, 5287050500, 26134330813, 132760735671, 692294900849, 3701754158688, 20275893222445, 113657560920970, 651449039159673, 3814790900995022, 22805438484189851

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..40
• M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967

Crossrefs

Cf. A008406.

Programs

Mathematica

Needs["Combinatorica`"]
Table[ NumberOfGraphs[n, n-1], {n, 1, 25}] (* Robert G. Wilson v *)
(* Second program (not needing Combinatorica): *)
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_, t_] := Product[Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]* v[[j]]/g]^g, {j, 1, i - 1}], {i, 2, Length[v]}]*Product[c = v[[i]]; t[c]^Quotient[c - 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
a[n_] := a[n] = Module[{s = 0}, Do[s += permcount[p]*edges[p, 1 + x^# &], {p, IntegerPartitions[n]}]; s/n!] // Expand // SeriesCoefficient[#, {x, 0, n-1}]&;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 35}] (* Jean-François Alcover, Aug 18 2022, after Andrew Howroyd in A008406 *)

Extensions

More terms from Vladeta Jovovic, Jan 15 2000