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A094199 Quadratic recurrence that arises when enumerating labeled connected graphs (called Wright's constants). 1
1, 49, 9800, 4412401, 3530881200, 4414129955298, 7945866428953600, 19467894010226044005, 62298157203907977632000, 252309651689367225339613486 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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.

FORMULA

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

EXAMPLE

a(2) = 2*(10-4)*(10-6)*a(1)+a(1) = 49 since a(1)=1.

MATHEMATICA

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-Fran├žois Alcover, Jun 20 2013 *)

CROSSREFS

Cf. A062980.

Sequence in context: A351598 A014801 A187406 * A194023 A195273 A222459

Adjacent sequences:  A094196 A094197 A094198 * A094200 A094201 A094202

KEYWORD

nonn

AUTHOR

Steven Finch, May 25 2004

STATUS

approved

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Last modified August 11 20:13 EDT 2022. Contains 356067 sequences. (Running on oeis4.)