%I #23 Sep 26 2018 02:54:01

%S 1,1,3,6,15,33,83,196,491,1214,3068,7754,19834,50872,131423,340763,

%T 887839,2321193,6090979,16031341,42319223,112003765,297164610,

%U 790190726,2105607907,5621642203,15036126167,40284850520,108102408101

%N Number of structurally isomeric homologs with molecular formula C_{3+n} H_{6+2n}.

%D Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).

%D G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987, p. 63.

%D Ching-Wan Lam, "Enumeration of isomers of alkylcyclopropanes by means of alkyl 1,1-biradicals", J. Math. Chem., 27 (2000), 23-25. [From _Parthasarathy Nambi_, Aug 24 2008]

%H Andrew Howroyd, <a href="/A063832/b063832.txt">Table of n, a(n) for n = 0..200</a>

%H G. Polya, <a href="https://projecteuclid.org/euclid.acta/1485888172">Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen</a>, Acta Math. 68 (1937), 145-254.

%F G.f.: A(x) = cycle_index(S3[S2]B(x)), where B(x) is g.f. for A000598.

%t G[n_] := Module[{g}, Do[g[x_] = 1 + x*(g[x]^3/6 + g[x^2]*g[x]/2 + g[x^3]/3) + O[x]^n // Normal, {n}]; g[x]];

%t T[n_, k_] := Module[{t = G[n], g}, t = x*((t^2 + (t /. x -> x^2))/2); g[e_] = (Normal[t + O[x]^Quotient[n, e]] /. x -> x^e) + O[x]^n // Normal; Coefficient[(Sum[EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[ k], g[1]*g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2])/2, x, n]];

%t a[n_] := T[n + 3, 3];

%t Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Jul 03 2018, after _Andrew Howroyd_ *)

%Y Column k=3 of A305059.

%Y Column 3 of a table (in Parks and Hendrickson) in which the subsequent columns are A116719, A120333, A120779, A120790, A120795, A121156, A121157.

%K nonn,nice

%O 0,3

%A _Vladeta Jovovic_, Aug 21 2001