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

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

Author

R. J. Mathar, May 10 2018

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274

Crossrefs

Cf. A054921 (row sums), A001349 (1st column), A126100 (2nd column), A303831 (3rd column), A294783 (trees).

Programs

PARI

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

Formula

T(n,k) = T(n,n-k).