Voozh

A054923

Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 5, 6, 0, 0, 0, 1, 5, 13, 11, 0, 0, 0, 0, 4, 19, 33, 23, 0, 0, 0, 0, 2, 22, 67, 89, 47, 0, 0, 0, 0, 1, 20, 107, 236, 240, 106, 0, 0, 0, 0, 1, 14, 132, 486, 797, 657, 235, 0, 0, 0, 0, 0, 9, 138, 814, 2075, 2678, 1806, 551, 0, 0, 0, 0, 0, 5, 126, 1169, 4495, 8548, 8833, 5026, 1301

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OFFSET

0,10

COMMENTS

The diagonal n = k+1 is A000055(n). - Jonathan Vos Post, Aug 10 2008

REFERENCES

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 93, Table 4.2.2; p. 241, Table A2.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1325

G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 57.

Gordon Royle, Small graphs

Gus Wiseman, Non-isomorphic representatives of the 12 connected graphs counted in row 5.

EXAMPLE

Triangle begins:

0, 1;

0, 0, 1;

0, 0, 1, 2;

0, 0, 0, 2, 3;

0, 0, 0, 1, 5 6;

0, 0, 0, 1, 5, 13, 11;

0, 0, 0, 0, 4, 19, 33, 23;

0, 0, 0, 0, 2, 22, 67, 89, 47;

0, 0, 0, 0, 1, 20, 107, 236, 240, 106;

... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 6 with 6 nodes). [Typo corrected by Anders Haglund, Jul 08 2008]

PROG

(PARI)

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i), 1))}

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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}

G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); s/n!}

T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}

{my(A=T(10)); for(n=1, #A, print(A[n, 1..n]))} \\ Andrew Howroyd, Oct 23 2019

CROSSREFS

Main diagonal is A000055.

Subsequent diagonals give the number of connected unlabeled graphs with n nodes and n+k edges for k=0..2: A001429, A001435, A001436.

Cf. A002905 (row sums), A001349 (column sums), A008406, A046751 (transpose), A054924 (transpose), A046742 (w/o left column), A343088 (labeled).

Cf. A000664, A007718, A050535, A054923, A191646, A191970, A317672, A322114, A322151.

Sequence in context: A284272 A175070 A343873 * A384850 A263145 A057108

Adjacent sequences: A054920 A054921 A054922 * A054924 A054925 A054926

KEYWORD

nonn,easy,nice,tabl