A005217 Number of unlabeled unit interval graphs with n nodes. (Formerly M1186)

1, 2, 4, 9, 21, 55, 151, 447, 1389, 4502, 15046, 51505, 179463, 634086, 2265014, 8163125, 29637903, 108282989, 397761507, 1468063369, 5441174511, 20242989728, 75566702558, 282959337159, 1062523000005, 4000108867555, 15095081362907, 57088782570433

Offset

1,2

References

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.

R. W. Robinson, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.

Links

R. W. Robinson, Table of n, a(n) for n = 1..190

Phil Hanlon, Counting interval graphs, Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.

Formula

G.f. A(x) = x + 2x^2 + 4x^3 + 9x^4 + 21x^5 + ... satisfies 1 + A(x) = exp( Sum_{k >= 1} psi(x^k)/k ), where psi(x) = (1+2*x-sqrt(1-4*x)*sqrt(1-4*x^2))/(4*sqrt(1-4*x^2)) is the g.f. for A007123.

For asymptotics, see for example Finch.

Mathematica

m = 30;
A[x_] = (-1 + Exp[Sum[psi[x^k]/k, {k, 1, m}]] /. psi[x_] -> (1 + 2 x - Sqrt[1 - 4 x] Sqrt[1 - 4 x^2])/(4 Sqrt[1 - 4 x^2])) + O[x]^m;
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Oct 24 2019 *)

Crossrefs

Sequence in context: A198304 A032129 A304914 * A148072 A001430 A148073

Adjacent sequences: A005214 A005215 A005216 * A005218 A005219 A005220

Keyword

nonn

Author

N. J. A. Sloane

Status

approved