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A381467 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k cycles and no node a member of more than one cycle, 0 <= k <= floor(n/3).

%I A381467 #16 Feb 25 2025 13:13:38
%S A381467 1,1,1,1,1,2,2,3,5,6,13,1,11,33,4,23,89,21,47,240,85,2,106,657,345,16,
%T A381467 235,1806,1289,109,551,5026,4713,627,6,1301,13999,16622,3259,64,3159,
%U A381467 39260,57535,15576,598,7741,110381,195212,69983,4394,18,19320,311465,653318,299354,28286,295
%N A381467 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k cycles and no node a member of more than one cycle, 0 <= k <= floor(n/3).
%C A381467 All such graphs are cactus graphs (with bridges allowed).
%H A381467 Andrew Howroyd, <a href="/A381467/b381467.txt">Table of n, a(n) for n = 0..1750</a> (rows 0..100)
%H A381467 R. J. Mathar, <a href="http://arxiv.org/abs/1808.06264">Counting connected graphs without overlapping cycles</a>, arXiv:1808.06264 [math.CO] (2018).
%H A381467 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A381467 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%F A381467 T(3*n, n) = A380634(n).
%e A381467 Triangle begins:
%e A381467      1;
%e A381467      1;
%e A381467      1;
%e A381467      1,     1;
%e A381467      2,     2;
%e A381467      3,     5;
%e A381467      6,    13,     1;
%e A381467     11,    33,     4;
%e A381467     23,    89,    21;
%e A381467     47,   240,    85,     2;
%e A381467    106,   657,   345,    16;
%e A381467    235,  1806,  1289,   109;
%e A381467    551,  5026,  4713,   627,   6;
%e A381467   1301, 13999, 16622,  3259,  64;
%e A381467   3159, 39260, 57535, 15576, 598;
%e A381467   ...
%o A381467 (PARI)
%o A381467 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 A381467 raise(p,d) = {my(n=serprec(p,x)-1); substvec(p + O(x^(n\d+1)), [x, y], [x^d,y^d])}
%o A381467 R(n,y)={my(g=x+O(x^2)); for(n=2, n, my(p=x*EulerMTS(g), p2=raise(p,2)); g=p + p*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
%o A381467 G(n,y=1)={my(g=R(n,y), p = x*EulerMTS(g) + O(x*x^n));
%o A381467   my( r=((1 + p)^2/(1 - raise(p,2)) - 1)/2 );
%o A381467   my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p,d))) );
%o A381467   1 + p + (raise(g,2) - g^2 + y*(r + c - 2*p - p^2 - raise(p,2)))/2 }
%o A381467 T(n)={[Vecrev(p) | p<-Vec(G(n,y))]}
%o A381467 {my(A=T(15)); for(i=1, #A, print(A[i]))}
%Y A381467 Row sums are A381468.
%Y A381467 Columns k=0..2 are A000055, A001429, A381470.
%Y A381467 Cf. A380631, A380634.
%K A381467 nonn,tabf
%O A381467 0,6
%A A381467 _Andrew Howroyd_, Feb 24 2025