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A005703 Number of n-node connected graphs with at most one cycle.
(Formerly M1151)
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) is the number of pseudotrees on n nodes. - Eric W. Weisstein, Jun 11 2012

REFERENCES

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).

LINKS

Washington Bomfim, Table of n, a(n) for n = 0..500

R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988

Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.

Eric Weisstein's World of Mathematics, Pseudotree

Wikipedia, Pseudoforest

FORMULA

a(n) = A000055(n) + A001429(n).

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A000055, A000081, A001429, A134964 (number of pseudoforests).

Sequence in context: A037444 A151526 A099526 * A172383 A003081 A100133

Adjacent sequences:  A005700 A005701 A005702 * A005704 A005705 A005706

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

More terms from Vladeta Jovovic, Apr 19 2000 and from Michael Somos, Apr 26 2000

a(27) corrected and a(28) and a(29) computed by Washington Bomfim, May 14 2008

STATUS

approved

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Last modified November 30 13:27 EST 2021. Contains 349419 sequences. (Running on oeis4.)