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URL: https://oeis.org/A005217

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A005217
Number of unlabeled unit interval graphs with n nodes.
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
OFFSET
1,2
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Phil Hanlon, Counting interval graphs, Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.
Sascha Kurz, Equivalent bounded confidence processes, arXiv:2512.18016 [physics.soc-ph], 2025. See p. 18, Tables 1 and 2.
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
KEYWORD
nonn
STATUS
approved