A094199 Quadratic recurrence that arises when enumerating labeled connected graphs (called Wright's constants).

1, 49, 9800, 4412401, 3530881200, 4414129955298, 7945866428953600, 19467894010226044005, 62298157203907977632000, 252309651689367225339613486

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-François 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