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A005703
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Number of n-node connected graphs with at most one cycle.
(Formerly M1151)
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29
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1, 1, 1, 2, 4, 8, 19, 44, 112, 287, 763, 2041, 5577, 15300, 42419, 118122, 330785, 929469, 2621272, 7411706, 21010378, 59682057, 169859257, 484234165, 1382567947, 3952860475, 11315775161, 32430737380, 93044797486, 267211342954, 768096496093, 2209772802169
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OFFSET
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0,4
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COMMENTS
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Also unlabeled connected graphs covering n vertices with at most n edges. For this definition we have a(1) = 0 and possibly a(0) = 0. - Gus Wiseman, Feb 20 2024
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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EXAMPLE
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Representatives of the a(0) = 1 through a(5) = 8 graphs:
{} . {12} {12,13} {12,13,14} {12,13,14,15}
{12,13,23} {12,13,24} {12,13,14,25}
{12,13,14,23} {12,13,24,35}
{12,13,24,34} {12,13,14,15,23}
{12,13,14,23,25}
{12,13,14,23,45}
{12,13,14,25,35}
{12,13,24,35,45}
(End)
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MATHEMATICA
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Needs["Combinatorica`"]; nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}];
a[0] = 0;
b = Drop[Flatten[
sol = SolveAlways[
0 == Series[
t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
CoefficientList[
Series[CycleIndex[DihedralGroup[n], s] /.
Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; CoefficientList[
Series[r[x] - (r[x]^2 - r[x^2])/2 + d[x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 17 2014 *)
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PROG
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(PARI) \\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + g(1) + (g(2) - g(1)^2)/2 + sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2)}; \\ Andrew Howroyd and Washington Bomfim, May 15 2021
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CROSSREFS
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A062734 counts connected graphs by number of edges.
Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, A368951, A369197, A370317, A370318.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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