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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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A128101 A211351 A124802

Adjacent sequences:  A305056 A305057 A305058 * A305060 A305061 A305062

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, May 24 2018

STATUS

approved

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Last modified December 1 16:55 EST 2020. Contains 338846 sequences. (Running on oeis4.)