A216234 Cumulated number of increasing admissible cuts of rooted plane trees of size n.

0, 1, 2, 8, 44, 312, 2772, 30024, 385688, 5737232, 96959396, 1834244296, 38390799592, 880648730416, 21968596282440, 592083291341520, 17144219069647920, 530774988154571040, 17495673315094986180, 611738880367145595720, 22614424027640541372360

Offset: 0

Author

Antoine Genitrini, Mar 14 2013

Keywords

• nonn

Comments

In concurrency theory, a(n) is also the cumulated sizes of computation trees induced by interleaved concurrent processes of size n.

References

• O. Bodini, A. Genitrini and F. Peschanski. Enumeration and Random Generation of Concurrent Computations. In proc. 23rd International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA'12), Discrete Mathematics and Theoretical Computer Science, pp 83-96, 2012.

Links

• O. Bodini, A. Genitrini, F. Peschanski, A Quantitative Study of Pure Parallel Processes, arXiv preprint arXiv:1407.1873, 2014

Crossrefs

Cf. A007852.

Programs

Mathematica

Flatten[{0,RecurrenceTable[{-16*(-3+n)*(-7+2*n)*(-5+2*n)*a[-3+n]+4*(-5+2*n)*(3-12*n+4*n^2)*a[-2+n]-2*(28-23*n+2*n^3)*a[-1+n]+(-2+n)*(-1+2*n)*a[n]==0,a[1]==1,a[2]==2,a[3]==8},a,{n,1,20}]}] (* Vaclav Kotesovec, Mar 08 2014 *)

Python

def a(n):
   if n < 3:
      return n
   l = [0,1,2]
   for i in range(n-2):
      l[i%3] = ( (16*i-64*i**3)*l[i%3]+(12+72*i+112*i**2+32*i**3)*l[(i+1)%3]+(-26-62*i-4*i**3-36*i**2)*l[(i+2)%3] ) / (-5-7*i-2*i**2)
   return l[i%3]

Formula

P-recurrence: (16*n-64*n^3)*a(n)+(12+72*n+112*n^2+32*n^3)*a(n+1)+(-26-62*n-4*n^3-36*n^2)*a(n+2)+(5+7*n+2*n^2)*a(n+3) = 0; a(0)=0; a(1)=1; a(2)=2.

a(n) ~ 2^(n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Mar 08 2014