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A006343
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Arkons: number of elementary maps with n-1 nodes.
(Formerly M3400)
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13
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1, 0, 1, 1, 4, 10, 34, 112, 398, 1443, 5387, 20482, 79177, 310102, 1228187, 4910413, 19792582, 80343445, 328159601, 1347699906, 5561774999, 23052871229, 95926831442, 400587408251, 1678251696379, 7051768702245, 29710764875014
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OFFSET
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0,5
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REFERENCES
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K. Appel and W. Haken, Every planar map is four colorable. With the collaboration of J. Koch. Contemporary Mathematics, 98. American Mathematical Society, Providence, RI, 1989. xvi+741 pp. ISBN: 0-8218-5103-9.
F. R. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture. Ph.D. Dissertation, Kansas State Univ., 1974.
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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a(n-1) = Sum (n-k-1)^(-1)*binomial(n, k)*binomial(2*n-3*k-4, n-2*k-2); k = 0..[ (n-2)/2 ], n >= 3.
O.g.f. A(x) equals 1/x * series reversion ( x/(1 + x^2*C(x) ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for A000108.
A(x) is an algebraic function satisfying x^3*A^3(x) - (x - 1)*A^2(x) + (x - 2)*A(x) + 1 = 0. (End)
a(n) ~ sqrt(s*(1 - s + 3*r^2*s^2) / (1 - r + 3*r^3*s)) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.2229935155751761877673240243525445951244491757706... and s = 1.116796494086474135831052534637944909439048671327... are real roots of the system of equations 1 + (r-2)*s + r^3*s^3 = (r-1)*s^2, r + 2*s + 3*r^3*s^2 = 2 + 2*r*s. - Vaclav Kotesovec, Nov 27 2017
Conjecture: D-finite with recurrence: -(n+3)*(n-1)*a(n) +(11*n^2-2*n-45)*a(n-1) -(37*n+29)*(n-3)*a(n-2) +(29*n^2-125*n+78)*a(n-3) +(61*n-106)*(n-3)*a(n-4) -155*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Feb 20 2020
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MAPLE
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MATHEMATICA
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a[n_] := Sum[ Binomial[n, k]*Binomial[2*n-3*k-4, n-2*k-2]/(n-k-1), {k, 0, (n-2)/2}]; a[0] = 1; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Dec 14 2012, from formula *)
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PROG
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(Haskell)
a006343 0 = 1
a006343 n = sum $ zipWith div
(zipWith (*) (map (a007318 n) ks)
(map (\k -> a007318 (2*n - 3*k - 4) (n - 2*k - 2)) ks))
(map (toInteger . (n - 1 -)) ks)
where ks = [0 .. (n - 2) `div` 2]
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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EXTENSIONS
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Erroneously duplicated term 4 removed per Frank Bernhart's report by Max Alekseyev, Feb 11 2010
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STATUS
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approved
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