A006082 revision #65
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A006082
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Number of labeled projective plane trees (or "flat" trees) with n nodes.
(Formerly M0798)
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8
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1, 1, 1, 2, 3, 6, 12, 27, 65, 175, 490, 1473, 4588, 14782, 48678, 163414, 555885, 1913334, 6646728, 23278989, 82100014, 291361744, 1039758962, 3729276257, 13437206032, 48620868106, 176611864312, 643834562075, 2354902813742, 8640039835974, 31791594259244
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OFFSET
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1,4
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COMMENTS
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Also, the number of noncrossing partitions up to rotation and reflection composed of n-1 blocks of size 2. - Andrew Howroyd, May 03 2018
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REFERENCES
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R. W. Robinson, personal communication.
R. W. Robinson, Efficiency of power series operations for graph counting, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
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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T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360. [The sequence is mentioned on page 355, but because of a miscalculation it is given, incorrectly, as 1, 1, 1, 2, 3, 6, 12, 25. Thanks to David Broadhurst for this information. - N. J. A. Sloane, Apr 06 2022]
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]
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FORMULA
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MATHEMATICA
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u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #]&] + DivisorSum[GCD[n-1, k], EulerPhi[#]*Binomial[n*k/#, (n-1)/#]&])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
a[n_] := T[n, 2];
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PROG
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{A006082(n)=my(c(n)=binomial(2*n, n));
if(n<2, 1, n--; (c(n)+if(n%2, 2*n*(n+2), (n+1)^2)*c(n\2)
+(n+1)*sumdiv(n, d, if(d>2, eulerphi(d)*c(n/d))))/(4*n*(n+1))); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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editing
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