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

0,3

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

Table of n, a(n) for n=0..20.

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

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

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 *)

Prog

(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]

Crossrefs

Cf. A007852.

Sequence in context: A212913 A321628 A005363 * A123307 A293905 A244430

Adjacent sequences: A216231 A216232 A216233 * A216235 A216236 A216237

Keyword

nonn

Author

Antoine Genitrini, Mar 14 2013

Status

approved