A288265 Triangle read by rows: T(n,k) is the number of labeled connected planar graphs on n vertices and k edges.
1, 1, 3, 1, 16, 15, 6, 1, 125, 222, 205, 120, 45, 10, 1296, 3660, 5700, 6165, 4935, 2937, 1125, 195, 16807, 68295, 156555, 258125, 330456, 334530, 254275, 131985, 40950, 5712, 262144, 1436568, 4483360, 10230360, 18528216, 27261192, 31761744, 27958920, 17666320, 7513632, 1922760, 223440, 4782969, 33779340, 136368414, 405918324, 970196283, 1910996136, 3058785990, 3866563764, 3754432899, 2724326136, 1425385584, 507370500, 109907280, 10929600
Offset: 1
Examples
A(x;t) = x + t*x^2/2! + (3*t^2 + t^3)*x^3/3! + (16*t^3 + 15*t^4 + 6*t^5 + t^6)*x^4/4! + ... Triangle starts: n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [1] 1; [2] 0 1; [3] 0, 0, 3, 1; [4] 0, 0, 0, 16, 15, 6, 1; [5] 0, 0, 0, 0, 125, 222, 205, 120, 45, 10; [6] 0, 0, 0, 0, 0, 1296, 3660, 5700, 6165, 4935, 2937, 1125, 195; [7] ...
Links
- Gheorghe Coserea, Rows n = 1..126, flattened
- E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
- M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
- Omer Gimenez, Marc Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309-329.
Programs
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PARI
Q(n,k) = { \\ c-nets with n-edges, k-vertices if (k < 2+(n+2)\3 || k > 2*n\3, return(0)); sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2* (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) - 4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1)))); }; A100960_ser(N) = { my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)), q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 2*n\3), k, Q(n,k)),'t))), d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1), g2=intformal(t^2/2*((1+d)/(1+x)-1))); serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n,'t),'x,'t)))*'x); }; A288265_ser(N) = { my(x='x+O('x^(N+3)), b = t*x^2/2 + serconvol(A100960_ser(N), exp(x)), g1=intformal(serreverse(x/exp(b'))/x)); serlaplace(g1); }; A288265_seq(N) = { my(v=Vec(A288265_ser(N))); vector(#v, n, Vecrev(v[n]/t^(n-1))); }; concat(A288265_seq(9))
Comments