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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, 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.
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
Sequence in context: A371758 A281482 A132889 * A149061 A149062 A066979
KEYWORD
nonn
AUTHOR
Ben Burns, Jul 22 2014
STATUS
approved