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A005217 Number of unlabeled unit interval graphs with n nodes.
(Formerly M1186)
2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified April 20 23:46 EDT 2021. Contains 343143 sequences. (Running on oeis4.)