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 A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1). 14
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 EXAMPLE Triangle begins:   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;   ... (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(sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i) )} 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 Cf. A002905 (row sums), A001349 (column sums), A008406, A046751 (transpose), A054924 (transpose), A046742 (w/o left column). Cf. A000664, A007718, A050535, A054923, A191646, A191970, A317672, A322114, A322151. Sequence in context: A275964 A284272 A175070 * A263145 A057108 A063958 Adjacent sequences:  A054920 A054921 A054922 * A054924 A054925 A054926 KEYWORD nonn,easy,nice,tabl AUTHOR EXTENSIONS a(83)-a(89) corrected by Andrew Howroyd, Oct 24 2019 STATUS approved

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Last modified January 29 07:03 EST 2020. Contains 331337 sequences. (Running on oeis4.)