A121162 - OEIS

%I #12 May 25 2018 03:23:52

%S 1,3,13,41,141,440,1391,4244,12913,38651,115082,339646,997709,2915010,

%T 8485573,24612666,71191458,205393819,591330506,1699226719,4874925420,

%U 13965498369,39957144189,114193222891,326023307022,929958622555,2650483647976,7548608038736

%N Number of separated bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

%C Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles of equal length without any shared node and all nodes having degree at most 4. - _Andrew Howroyd_, May 25 2018

%H Andrew Howroyd, <a href="/A121162/b121162.txt">Table of n, a(n) for n = 6..200</a>

%H Camden A. Parks and James B. Hendrickson, <a href="https://doi.org/10.1021/ci00002a021">Enumeration of monocyclic and bicyclic carbon skeletons</a>, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).

%o (PARI) \\ here G is A000598 as series

%o 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}

%o C1(n)={sum(k=1, n\4, d1^(4*k) + 2*d1^(2*k)*d2^k + d2^(2*k))*(1 + d1^2)/(8*(1-d1))}

%o C2(n)={sum(k=1, n\4,  2*(d2^(2*k) + d4^k)*(1 + d2))*(1+d1)/(8*(1-d2))}

%o seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2],[g(d,1),g(d,2)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d4], [g(d,1),g(d,2),g(d,4)]))} \\ _Andrew Howroyd_, May 25 2018

%Y Cf. A121158, A125669.

%K nonn

%O 6,2

%A _Parthasarathy Nambi_, Aug 13 2006

%E More terms from _N. J. A. Sloane_, Aug 27 2006

%E Terms a(26) and beyond from _Andrew Howroyd_, May 25 2018