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A370064 Triangle read by rows: T(n,k) is the number of simple connected graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

%I A370064 #10 Feb 25 2024 21:05:13
%S A370064 1,1,0,1,0,0,1,3,0,0,10,16,12,0,0,238,250,180,60,0,0,11368,8496,4560,
%T A370064 1920,360,0,0,1014888,540568,211680,75600,21000,2520,0,0,166537616,
%U A370064 61672192,17186624,4663680,1226400,241920,20160,0,0,50680432112,12608406288,2416430016,469336896,98431200,20109600,2963520,181440,0,0
%N A370064 Triangle read by rows: T(n,k) is the number of simple connected graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).
%H A370064 Andrew Howroyd, <a href="/A370064/b370064.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A370064 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 A370064 T(n, n-2) = n!/2 = A001710(n) for n >= 2.
%e A370064 Triangle begins:
%e A370064         1;
%e A370064         1,      0;
%e A370064         1,      0,      0;
%e A370064         1,      3,      0,     0;
%e A370064        10,     16,     12,     0,     0;
%e A370064       238,    250,    180,    60,     0,    0;
%e A370064     11368,   8496,   4560,  1920,   360,    0, 0;
%e A370064   1014888, 540568, 211680, 75600, 21000, 2520, 0, 0;
%e A370064   ...
%o A370064 (PARI)
%o A370064 J(p, n)={my(u=Vecrev(p,1+n)); forstep(k=n, 1, -1, u[k] -= k*u[k+1]; u[k]/=n+1-k); u}
%o A370064 G(n)={log(x/serreverse(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))))}
%o A370064 T(n)={my(v=Vec(serlaplace( 1 + ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ))); vector(#v, n, J(v[n], n-1))}
%o A370064 { my(A=T(7)); for(i=1, #A, print(A[i])) }
%Y A370064 Columns k=0..3 are A013922(n>1), A013923, A013924, A013925.
%Y A370064 Row sums are A001187.
%Y A370064 Cf. A001710, A325111 (unlabeled version).
%K A370064 nonn,tabl
%O A370064 0,8
%A A370064 _Andrew Howroyd_, Feb 23 2024