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
Examples
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;
...
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
Crossrefs
Programs
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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 *) -
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)}
Comments