The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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-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 * A128101 A211351 A124802 Adjacent sequences:  A305056 A305057 A305058 * A305060 A305061 A305062 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, May 24 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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