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%I #19 Jan 25 2021 20:17:16
%S 1,1,0,0,0,1,1,0,1,0,3,1,1,1,0,11,5,3,1,1,0,61,29,14,5,2,1,0,507,224,
%T 86,25,8,2,1,0,7442,2666,762,184,48,11,3,1,0,197772,50779,10173,1890,
%U 374,72,16,3,1,0,9808209,1653431,220627,29252,4252,660,115,20,4,1,0
%N Triangle T(n,k) read by rows: number of simple connected graphs with n nodes and k endpoints, n >= 0, 0 <= k <= n.
%C Endpoints are vertices with 0 or 1 (less than 2) edges.
%H Andrew Howroyd, <a href="/A304222/b304222.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e The triangle starts in row n=0 with column 0 <= k <= n as:
%e 1;
%e 1, 0;
%e 0, 0, 1;
%e 1, 0, 1, 0;
%e 3, 1, 1, 1, 0;
%e 11, 5, 3, 1, 1, 0;
%e 61, 29, 14, 5, 2, 1, 0;
%e 507, 224, 86, 25, 8, 2, 1, 0;
%e 7442, 2666, 762, 184, 48, 11, 3, 1, 0;
%e 197772, 50779, 10173, 1890, 374, 72, 16, 3, 1, 0;
%o (PARI)
%o InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i) )}
%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o G(n)={sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * prod(i=1, #p, (1 - x^p[i])/(1 - (x*y)^p[i]) + O(x*x^(n-k)))); x^k*s/k!)}
%o T(n)={my(v=InvEulerMT(Vec(G(n)-1))); v[2]=y^2; concat([[1]], vector(#v, n, Vecrev(v[n], n+1))) }
%o my(A=T(10)); for(n=1, #A, print(A[n])) \\ _Andrew Howroyd_, Jan 22 2021
%Y Cf. A001349 (row sums), A004108 (first column), A055290 (trees only), A327371.
%K nonn,tabl
%O 0,11
%A _R. J. Mathar_, May 11 2018
%E Terms a(55) and beyond from _Andrew Howroyd_, Jan 22 2021
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