

A054026


a(n) is the number of sets of natural numbers [a,b,c,d,e] that can be produced with the numbers [0..n] such that the values of all the distinct parenthesized expressions of abcde are different.


1



0, 0, 0, 0, 300, 1296, 4116, 9984, 21384, 40800, 72600, 120960, 192660, 294000, 434700, 623616, 873936, 1197504, 1611504, 2131200, 2778300, 3571920, 4538820, 5702400, 7095000, 8744736, 10690056, 12964224, 15612324, 18673200, 22199100, 26234880, 30840480, 36067200, 41983200, 48646656, 56134476
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OFFSET

0,5


COMMENTS

There are 14 ways to put parentheses in the expression a  b  c  d  e: ((a  (b  c))  d)  e, (((a  b)  c)  d)  e, ((a  b)  (c  d))  e, etc. This sequence describes how many sets of natural numbers [a,b,c,d,e] can be produced with the numbers {0,1,2,3,...,n} such that the values of all the distinct expressions are different.
It can be shown that in the set of expressions obtained this way, for any number of variables, a is always positive, b is always negative, and the other variables appear with every possible combination of signs. Therefore, the valid ktuples of numbers in [0..n] are precisely those such that every subset of {c,d,e,...}, including the empty subset, has a distinct sum. For 5 variables, there are n*(n1)*(n2) ways to choose distinct, nonzero values for c, d, and e. For each k, there are floor((n1)/2) ways to choose distinct numbers x and y in [0..n] such that x + y = k. Summing over all k in [0..n], allowing arbitrary permutations of {x,y,k}, and allowing a and b to be any value gives the formula below.  Charlie Neder, Jan 13 2019


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FORMULA



EXAMPLE

For example, no such sets can be produced with only 0's, only 0's and 1's, only 0's and 1's and 2's, only 1's and 2's and 3's; with {0,1,2,3,4}, 300 such sets can be produced.


PROG

(PARI) a(n) = (1+n)^2*(3*(1)^n+4*n^318*n^2+20*n3)/4; \\ Jinyuan Wang, Jun 27 2020


CROSSREFS

Cf. A045991 (similar for a  b  c), A047929 (similar for a  b  c  d).


KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS



STATUS

approved



