A245390 Number of ways to build an expression using non-redundant parentheses in an expression of n variables, counted in decreasing sub-partitions of equal size.

1, 1, 3, 11, 39, 145, 514, 1901, 6741, 24880, 87791, 325634, 1152725, 4251234, 15052387, 55750323, 197031808, 729360141, 2579285955, 9539017676, 33822222227, 124889799518, 440743675148, 1635528366655, 5790967247863, 21360573026444, 75643815954959, 280096917425535

Offset

1,3

Comments

Expressions contain only binary operators. Parentheses around a single variable or the entire expression are considered redundant or unnecessary.

If the number of partitions (parenthetical group) does not evenly divide the number of variables, combinations of remaining variables are counted but not divided into sub-partitions.

Links

Ben Burns, Table of n, a(n) for n = 1..98

Example

For n=3, the a(3)=3 different possibilities are
1)  x+x+x,
2)  (x+x)+x,
3)  x+(x+x).
For n=4, the a(4)=11 different possibilities are
1)  x+x+x+x,
2)  (x+x+x)+x,
3)  ((x+x)+x)+x,
4)  (x+(x+x))+x,
5)  x+(x+x+x),
6)  x+((x+x)+x),
7)  x+(x+(x+x)),
8)  (x+x)+x+x,
9)  x+(x+x)+x,
10) x+x+(x+x),
11) (x+x)+(x+x).
For n=5, a(5)=39 is the first time this sequence differs from A001003. The combinations for (x+x+x)+(x+x) and (x+x)+(x+x+x) are not counted as these two partitions are not of equal size.

Prog

(Python)
import math
import operator as op
def binom(n, r):
....r = min(r, n-r)
....if r == 0: return 1
....numer = reduce(op.mul, range(n, n-r, -1))
....denom = reduce(op.mul, range(1, r+1))
....return numer//denom
max = 100
seq = [0, 1, 1]
for n in range(3, max) :
....sum = 0
....for choose in range(2, n+1):
........f = math.ceil(n/choose)
........f+=1
........for times in range(1, int(f)) :
............if choose*times > n :
................continue
............if choose==n and times==1 :
................sum += 1
............else :
................sum += binom(n - (choose*times) + times, times) * (seq[choose]**times)
....seq.append(sum)
for (i, x) in enumerate(seq) :
....if i==0:
........continue
....print i, x

Crossrefs

Cf. A001003

Sequence in context: A371758 A281482 A132889 * A149061 A149062 A066979

Adjacent sequences: A245387 A245388 A245389 * A245391 A245392 A245393

Keyword

nonn

Author

Ben Burns, Jul 22 2014

Status

approved