A curious binomial identity

DISCRETE MATHEMATICS ELSEVIER

Discrete

Mathematics

131 (1994) 335-337

Communication

A curious binomial identity Neil J. Calkin School qf Mathematics,

Georgia Institute

of Technology,

Atlanta,

GA 30332, USA

Communicated by G.E. Andrews Received 22 October 1993

Abstract In this communication binomial

we shall prove a curious identity of sums of powers of the partial sum of

coefficients.

1. An identity Theorem.

C;=O (~~_o(~))3=n23”-1+23”-~2”(~).

Proof. Definef,=C;=,

(Ci=O(k”))3. It is sufficient

to show that

Write A,=C:=,(,“). Then fn=C;=oA:.

0012-365X/94/%07.00 Q 1994-El SSDI 0012-365X(92)00575-A

sevier Science B.V. All rights reserved

336

N.J. Calkin/ Discrete Mathematics 131 (1994) 335’337

L+I

-8f,=2

3n+‘+~~~*-(;))l-(2A,)3

=23n+3 -~12A:(;)+~6Al(~~-~(q)3. I=0

Observation 1. Cf=oAI(~)“=$2”(~)+~C;=o(~)3.

Indeed,

~Al(;)‘=~An-l(.1,)‘=~An-l(3 and since Al + Anel =2”+(T) we have

gA(lly=g(2”+(;))

(1)

=1$2n(;)2+lg)3

=;2f’,“)+;&(;)‘.

Observation 2. CyzoAf(l)=

iP3 f2 12”(z,n)+&=o(~)3

Indeed, 2”“=A.‘=~A;-A:_1=~A:I=0

=~3A:(;)-~3A1(;)2+~(1)

=~3A~(;)-~2n(~)-~~(~~. Hence 2 A:(;)=T+f2f3+$(7. I=0

A,I=0

(

‘I 0)

3

NJ.

Putting

these together,

CalkinlDiscrete

we indeed

f,+1-8f,=4.23”-3.22”

Mathematics

131 (1994) 335-337

331

find that

‘,” 0

as required.

0

2. An application In this section we shall discuss an application of this to order statistics. Observe that the expected value of the maximum of three independent Bernoulli random variables B(n, 3) is

Hence, by the central limit theorem, independent normal N(0, 1) random

the expected value m3 of the maximum variables is

subtracting off the mean, dividing by the standard formula for the asymptotics of n!

deviation

and applying

of three

Stirling’s