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A370065 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

%I A370065 #7 Feb 26 2024 01:21:13
%S A370065 1,1,0,2,0,0,5,3,0,0,24,28,12,0,0,334,390,240,60,0,0,13262,10776,6090,
%T A370065 2280,360,0,0,1106862,615860,255570,92820,23520,2520,0,0,175376048,
%U A370065 66625504,19275424,5446560,1429680,262080,20160,0,0,52257938968,13210716600,2592577512,520122456,112145040,22649760,3144960,181440,0,0
%N A370065 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).
%H A370065 Andrew Howroyd, <a href="/A370065/b370065.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A370065 S. Selkow, <a href="https://doi.org/10.1016/S0012-365X(97)00170-2">The enumeration of labeled graphs by number of cutpoints</a>, Discr. Math. 185 (1998), 183-191.
%F A370065 Exponential transform of A370064.
%F A370065 T(n+2, n) = A188588(n + 1).
%e A370065 Triangle begins:
%e A370065         1;
%e A370065         1,      0;
%e A370065         2,      0,      0;
%e A370065         5,      3,      0,     0;
%e A370065        24,     28,     12,     0,     0;
%e A370065       334,    390,    240,    60,     0,    0;
%e A370065     13262,  10776,   6090,  2280,   360,    0, 0;
%e A370065   1106862, 615860, 255570, 92820, 23520, 2520, 0, 0;
%e A370065   ...
%o A370065 (PARI) \\ Needs G, J defined in A370064.
%o A370065 T(n)={my(v=Vec( ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ), w=Vec(serlaplace(exp(sum(k=1, n, Polrev(J(v[k],k),y)*x^k, O(x*x^n)) )))); vector(#w, n, Vecrev(w[n],n))}
%o A370065 { my(A=T(8)); for(i=1, #A, print(A[i])) }
%Y A370065 Row sums are A006125.
%Y A370065 Column k=0 is A370066.
%Y A370065 Cf. A188588, A370064 (connected).
%K A370065 nonn,tabl
%O A370065 0,4
%A A370065 _Andrew Howroyd_, Feb 25 2024