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

0,7

Links

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

Formula

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

Example

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;

Prog

(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

Crossrefs

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

Sequence in context: A328484 A319442 A299772 * A175393 A184829 A338307

Adjacent sequences: A304308 A304309 A304310 * A304312 A304313 A304314

Keyword

nonn,tabl

Author

R. J. Mathar, May 10 2018

Status

approved