A121156 -id:A121156 - cp's OEIS frontend

A305059 Triangle read by rows: T(n,k) is the number of connected unicyclic graphs on n unlabeled nodes with cycle length k and all nodes having degree at most 4.

1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 15, 8, 4, 1, 1, 33, 24, 9, 5, 1, 1, 83, 55, 28, 12, 5, 1, 1, 196, 147, 71, 40, 13, 6, 1, 1, 491, 365, 198, 106, 47, 16, 6, 1, 1, 1214, 954, 521, 317, 136, 63, 18, 7, 1, 1, 3068, 2431, 1418, 868, 428, 190, 73, 21, 7, 1, 1

Offset: 3

Andrew Howroyd, May 24 2018

Equivalently, the number of monocyclic skeletons with n carbon atoms and a ring size of k.

			Triangle begins:
     1;
     1,   1;
     3,   1,   1;
     6,   4,   1,   1;
    15,   8,   4,   1,   1;
    33,  24,   9,   5,   1,  1;
    83,  55,  28,  12,   5,  1,  1;
   196, 147,  71,  40,  13,  6,  1, 1;
   491, 365, 198, 106,  47, 16,  6, 1, 1;
  1214, 954, 521, 317, 136, 63, 18, 7, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..1178 (rows n = 3..50)
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
Eric Weisstein's World of Mathematics, Unicyclic Graph

Columns 3..10 are A063832, A116719, A120333, A120779, A120790, A120795, A121156, A121157.
Row sums are A036671.
Cf. A000598.

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[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]];
Table[T[n, k], {n, 3, 13}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

\\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
T(n,k)={my(t=G(n)); t=x*(t^2+subst(t, x, x^2))/2; my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); polcoeff((sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2, n)}

A063832 Number of structurally isomeric homologs with molecular formula C_{3+n} H_{6+2n}.

Original entry on oeis.org

1, 1, 3, 6, 15, 33, 83, 196, 491, 1214, 3068, 7754, 19834, 50872, 131423, 340763, 887839, 2321193, 6090979, 16031341, 42319223, 112003765, 297164610, 790190726, 2105607907, 5621642203, 15036126167, 40284850520, 108102408101

Offset: 0

Vladeta Jovovic, Aug 21 2001

Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987, p. 63.
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]

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Polya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 145-254.

Column k=3 of A305059.

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

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[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]];
a[n_] := T[n + 3, 3];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

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

cp's OEIS Frontend

A305059 Triangle read by rows: T(n,k) is the number of connected unicyclic graphs on n unlabeled nodes with cycle length k and all nodes having degree at most 4.

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