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A380633 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes of degree at most 3 with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).

%I A380633 #9 Feb 24 2025 16:32:26
%S A380633 1,0,0,0,1,0,1,0,1,0,1,1,0,1,1,0,1,2,0,1,2,1,0,1,3,3,0,1,3,6,0,1,4,11,
%T A380633 2,0,1,4,17,5,0,1,5,26,17,0,1,5,36,37,2,0,1,6,50,78,12,0,1,6,65,140,
%U A380633 44,0,1,7,85,248,131,4,0,1,7,106,396,325,23
%N A380633 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes of degree at most 3 with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).
%C A380633 All such graphs are cactus graphs (with bridges allowed).
%H A380633 Andrew Howroyd, <a href="/A380633/b380633.txt">Table of n, a(n) for n = 0..1750</a> (rows 0..100)
%H A380633 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A380633 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%F A380633 T(3*n,n) = A000672(n).
%e A380633 Triangle begins:
%e A380633   1;
%e A380633   0;
%e A380633   0;
%e A380633   0, 1;
%e A380633   0, 1;
%e A380633   0, 1;
%e A380633   0, 1, 1;
%e A380633   0, 1, 1;
%e A380633   0, 1, 2;
%e A380633   0, 1, 2,  1;
%e A380633   0, 1, 3,  3;
%e A380633   0, 1, 3,  6;
%e A380633   0, 1, 4, 11,  2;
%e A380633   0, 1, 4, 17,  5;
%e A380633   0, 1, 5, 26, 17;
%e A380633   0, 1, 5, 36, 37, 2;
%e A380633   ...
%o A380633 (PARI)
%o A380633 raise(p,d) = {my(n=serprec(p,x)-1); substvec(p + O(x^(n\d+1)), [x, y], [x^d,y^d])}
%o A380633 R(n,y)={my(g=O(x^3)); for(n=1, (n-1)\2, my(p=x*(1 + g), p2=raise(p,2)); g=x*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
%o A380633 G(n,y=1)={my(g=R(n,y), p = x*(1+g) + O(x*x^n));
%o A380633   my( r=((1 + p)^2/(1 - raise(p,2)) - 1)/2 );
%o A380633   my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p,d))) );
%o A380633   1 + (raise(g,2) - g^2 + y*(r + c - 2*p - p^2 - raise(p,2)))/2 }
%o A380633 T(n)={[Vecrev(p) | p<-Vec(G(n,y))]}
%o A380633 {my(A=T(15)); for(i=1, #A, print(A[i]))}
%Y A380633 Columns 0..3 are A000007, A000012(n+3), A004526(n+4), A003453(n+4).
%Y A380633 Row sums are A380805.
%Y A380633 Cf. A000672, A380631 (with vertices of any degree).
%K A380633 nonn,tabf
%O A380633 0,18
%A A380633 _Andrew Howroyd_, Feb 24 2025