A303831
Birooted graphs: number of unlabeled connected graphs with n nodes rooted at 2 indistinguishable roots.
Original entry on oeis.org
0, 1, 3, 16, 98, 879, 11260, 230505, 7949596, 483572280, 53011686200, 10589943940654, 3880959679322754, 2623201177625659987, 3286005731275218388682, 7663042204550840483139108, 33407704152242477510352455230, 273327599183687887638526170380380
Offset: 1
A304311
Triangle T(n,k) read by rows: number of bicolored connected graphs with n nodes and k nodes of the first color.
Original entry on oeis.org
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, 879, 407, 112, 853, 4306, 11260, 17578, 17578, 11260, 4306, 853, 11117, 72489, 230505, 436371, 537272, 436371, 230505, 72489, 11117
Offset: 0
Triangle begins
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, 879, 407, 112;
853, 4306, 11260, 17578, 17578, 11260, 4306, 853;
11117, 72489, 230505, 436371, 537272, 436371, 230505, 72489, 11117;
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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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
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!}
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) )}
{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
A339039
Number of unlabeled connected simple graphs with n edges rooted at one distinguished vertex.
Original entry on oeis.org
1, 1, 2, 5, 13, 37, 114, 367, 1248, 4446, 16526, 63914, 256642, 1067388, 4590201, 20376849, 93240065, 439190047, 2126970482, 10579017047, 53983000003, 282345671127, 1512273916781, 8287870474339, 46438619162441, 265840311066579
Offset: 0
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\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1])/G(2*n, x+A, []))}
A126201
Number of rooted connected unlabeled planar graphs on n nodes.
Original entry on oeis.org
1, 1, 3, 11, 57, 375, 3398, 40043, 585440, 9895493
Offset: 1
A126101
Number of connected unlabeled graphs on n nodes that are rooted at a non-cut node.
Original entry on oeis.org
1, 1, 1, 2, 8, 44, 333, 3771, 67141, 2027119, 108880264, 10682138680, 1933264826485, 648235902085512, 404043306773404163, 469727521267710406698, 1022090075330054063050850, 4176738163895992397728030132, 32159402671814249205978139454278, 467987765188007308268883267776373304
Offset: 0
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\\ See A126100 for g; InvEulerT takes inverse Euler transform.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={concat([1, 1], InvEulerT(Vec(-1+Ser(vector(n, n, g(n-1, 1)))/Ser(vector(n, n, g(n-1, 0))))))} \\ Andrew Howroyd, Nov 23 2020
a(15)-a(18) corrected and terms a(20) and beyond from
Andrew Howroyd, Nov 23 2020
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