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
S. R. Finch, Shapes of binary trees, June 24, 2004. [Cached copy, with permission of the author]
S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
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.
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�ois Alcover, Jun 20 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Steven Finch, May 25 2004
EXTENSIONS
Name corrected by Steven Finch, Aug 12 2024
STATUS
approved