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

3,4

Comments

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

Links

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

Example

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;
  ...

Mathematica

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 *)

Prog

(PARI) \\ 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)}

Crossrefs

Columns 3..10 are A063832, A116719, A120333, A120779, A120790, A120795, A121156, A121157.

Row sums are A036671.

Cf. A000598.

Sequence in context: A125230 A208334 A162430 * A355996 A128101 A211351

Adjacent sequences: A305056 A305057 A305058 * A305060 A305061 A305062

Keyword

nonn,tabl

Author

Andrew Howroyd, May 24 2018

Status

approved