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A380631 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).

%I A380631 #15 Feb 25 2025 13:13:27
%S A380631 1,0,0,0,1,0,1,0,1,0,1,1,0,1,1,0,1,2,0,1,2,2,0,1,3,5,0,1,3,10,0,1,4,
%T A380631 17,6,0,1,4,26,18,0,1,5,38,51,0,1,5,52,106,18,0,1,6,70,205,87,0,1,6,
%U A380631 90,350,286,0,1,7,115,579,741,66,0,1,7,142,887,1660,406
%N A380631 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).
%C A380631 All such graphs are cactus graphs (with bridges allowed).
%H A380631 Andrew Howroyd, <a href="/A380631/b380631.txt">Table of n, a(n) for n = 0..1750</a> (rows 0..100)
%H A380631 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A380631 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%F A380631 T(3*n, n) = A380634(n).
%e A380631 Triangle begins:
%e A380631   1;
%e A380631   0;
%e A380631   0;
%e A380631   0, 1;
%e A380631   0, 1;
%e A380631   0, 1;
%e A380631   0, 1, 1;
%e A380631   0, 1, 1;
%e A380631   0, 1, 2;
%e A380631   0, 1, 2,  2;
%e A380631   0, 1, 3,  5;
%e A380631   0, 1, 3, 10;
%e A380631   0, 1, 4, 17,   6;
%e A380631   0, 1, 4, 26,  18;
%e A380631   0, 1, 5, 38,  51;
%e A380631   0, 1, 5, 52, 106, 18;
%e A380631   ...
%o A380631 (PARI)
%o A380631 EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
%o A380631 raise(p,d) = {my(n=serprec(p,x)-1); substvec(p + O(x^(n\d+1)), [x,y], [x^d,y^d])}
%o A380631 R(n,y)={my(g = O(x^3)); for(n=1, (n-1)\2, my(p=x*EulerMTS(g), p2=raise(p,2)); g=p*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
%o A380631 G(n,y=1)={my(g=R(n,y), p = x*EulerMTS(g) + O(x*x^n));
%o A380631   my( r=((1 + p)^2/(1 - raise(p,2)) - 1)/2 );
%o A380631   my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p,d))) );
%o A380631   1 + (raise(g,2) - g^2 + y*(r + c - 2*p - p^2 - raise(p,2)))/2 }
%o A380631 T(n)={[Vecrev(p) | p<-Vec(G(n,y))]}
%o A380631 { my(A=T(15)); for(i=1, #A, print(A[i])) }
%Y A380631 Columns 0..2 are A000007, A000012(n+3), A008619(n+6).
%Y A380631 Row sums are A380632.
%Y A380631 Cf. A380633, A380634, A381467.
%K A380631 nonn,tabf
%O A380631 0,18
%A A380631 _Andrew Howroyd_, Feb 24 2025