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A094199
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Quadratic recurrence that arises when enumerating labeled connected graphs (called Wright's constants).
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1
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OFFSET
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1,2
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COMMENTS
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The unknown constant in the article "Shapes of binary trees" by S. Finch (page 3, unsolved problem) is C = 0.0196207628432398766811334785902747944894235476341... = sqrt(15)/(20*Pi^2). - Vaclav Kotesovec, Jan 19 2015
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LINKS
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Table of n, a(n) for n=1..10.
S. Janson, The Wiener index of simply generated random trees, Random Structures Algorithms 22 (2003) 337-358.
S. Janson and P. Chassaing, The center of mass of the ISE and the Wiener index of trees, arXiv:math/0309284 [math.PR], 2003.
S. R. Finch, Shapes of binary trees, June 24, 2004. [Cached copy, with permission of the author]
E. M. Wright, The Number of Connected Sparsely Edged Graphs, Journal of Graph Theory Vol. 1 (1977), 317-330.
Jian Zhou, On a Mean Field Theory of Topological 2D Gravity, arXiv:1503.08546 [math.AG], 30 Mar 2015.
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FORMULA
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With a(0) = -1/2 one has for n > 0 the recurrence a(n) = 2*(5*n-4)*(5*n-6)*a(n-1)+sum(a(k)*a(n-k), k=1..n-1).
a(n) ~ sqrt(3) * 2^(n-1) * 5^(2*n-1/2) * n^(2*n-1) / (Pi * exp(2*n)). The unknown constant in theorem 4.2. in the article by S. Janson and P. Chassaing is beta = 5*sqrt(15)/(2*Pi^2) = 0.981038142161993834... . - Vaclav Kotesovec, Jan 19 2015
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EXAMPLE
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a(2) = 2*(10-4)*(10-6)*a(1)+a(1) = 49 since a(1)=1.
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = 2*(5*n - 4)*(5*n - 6)*a[n - 1] + Sum[a[k]*a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Jun 20 2013 *)
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CROSSREFS
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Cf. A062980.
Sequence in context: A351598 A014801 A187406 * A194023 A195273 A222459
Adjacent sequences: A094196 A094197 A094198 * A094200 A094201 A094202
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KEYWORD
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nonn
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AUTHOR
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Steven Finch, May 25 2004
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STATUS
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approved
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