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A304311 Triangle T(n,k) read by rows: number of bicolored connected graphs with n nodes and k nodes of the first color.

%I A304311 #10 May 14 2018 16:57:01
%S A304311 1,1,1,1,1,1,2,3,3,2,6,11,16,11,6,21,58,98,98,58,21,112,407,879,1087,
%T A304311 879,407,112,853,4306,11260,17578,17578,11260,4306,853,11117,72489,
%U A304311 230505,436371,537272,436371,230505,72489,11117
%N A304311 Triangle T(n,k) read by rows: number of bicolored connected graphs with n nodes and k nodes of the first color.
%H A304311 Andrew Howroyd, <a href="/A304311/b304311.txt">Table of n, a(n) for n = 0..1274</a>
%F A304311 T(n,k) = T(n,n-k).
%e A304311 Triangle begins
%e A304311       1;
%e A304311       1,     1;
%e A304311       1,     1,      1;
%e A304311       2,     3,      3,      2;
%e A304311       6,    11,     16,     11,      6;
%e A304311      21,    58,     98,     98,     58,     21;
%e A304311     112,   407,    879,   1087,    879,    407,    112;
%e A304311     853,  4306,  11260,  17578,  17578,  11260,   4306,   853;
%e A304311   11117, 72489, 230505, 436371, 537272, 436371, 230505, 72489, 11117;
%o A304311 (PARI)
%o A304311 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 A304311 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o A304311 S(n,y)={my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)*prod(i=1,#p,1+y^p[i])); s/n!}
%o A304311 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 A304311 {my(A=InvEulerMT(vector(10, n, S(n,y)))); for(n=0, #A, for(k=0, n, print1(polcoeff(if(n,A[n],1), k), ", ")); print)} \\ _Andrew Howroyd_, May 13 2018
%Y A304311 Cf. A054921 (row sums), A001349 (1st column), A126100 (2nd column), A303831 (3rd column), A294783 (trees).
%K A304311 nonn,tabl
%O A304311 0,7
%A A304311 _R. J. Mathar_, May 10 2018