American Journal of Computational Mathematics, 2013, 3, 31-37
doi:10.4236/ajcm.2013.33B006 Published Online September 2013 (http://www.scirp.org/journal/ajcm)
The Series of Reciprocals of Non-central Binomial
Coefficients
Laiping Zhang, Wanhui Ji
Yinchan Energy College, Yinchuan, China
Email: zhanglaiping79@163.com, jiwanhui2008@163.com
Received 2013
ABSTRACT
Utilizing Gamma-Beta function, we can build one series involving reciprocal o f non-central binomial coefficien ts, then
We can structure several new series of reciprocals of non-central binomial coefficients by item splitting , these new cre-
ated denominator of series contain 1 to 4 odd factors of binomial coefficients. As the result of splitting items, some
identities of series of numbers values of reciprocals of binomial coefficients are given. The method of splitting terms
offered in this paper is a new combinatorial analysis way and elementary method to create new series.
Keywords: Binomial Coefficients; Split Terms; Reciprocals; Series; Non-central; Closed Form
1. Introduction
Binomial coefficient series plays an important role among
Number theory, Graph Theory, mathematical statistics
and Probability Theory, etc mathematics branches. Bi-
nomial coefficient series conversion also plays key role
in research areas of combinatorial mathematics and
mathematical analysis, it aroused great attention and can
be referred to a large number of literature [1-7]. Paper
[2-5] the author used is called Lehmer series identity

2
22
1
(2 )2arcsin, ||1,
1
n
n
nn
xxx
x
nx

Some authors differential, integral, generating function,
the white to Beta-gamma function, hyper geometric
function series, recurrence and other mathematical tools
to get important results on series of reciprocals of bino-
mial coefficients. Paper [8] obtains alternating series
involving reciprocals of central binomial coefficients.
Paper [9] obtains series involving reciprocals of central
binomial coefficients. We did the research the series of
reciprocal of non-central binomial coefficient. Utilizing
Gamma-beta function, we obtain one series involving
reciprocal of non-central binomial coefficients. We can
structure several new series of reciprocal of non-central
binomial coefficients by splitting item; the new created
denominators of series contain 1 to 4 odd factors of
non-central binomial coefficients.
By continuously using the method of splitting items,
we can get the series involving reciprocal of non-central.
Binomial coefficients of which denominator contain
5odd factor, 6 odd factor, … and p odd factor.
Therefore splitting term method in this paper is an
elementary to build new series and also provide a new
combinatorial analysis method. As the result of analysis
some identities of series of number values of reciprocal
of non-central binomial coefficients also provided in this
paper.
2. Main Results and Proof of Theorem
Theorem 1 one series of reciprocals of non-central bi-
nomial coefficients

21
0
22
2
22
=
28
arctan,0 4
4(4) 44
24 4
ln, 40
4(4) 44
k
k
kk
xD
xx
xxxx xx
xx xx
xxxx xxx

 
 
 
 
(0)
Proof From power series theories easy to understand,
the series
 
21 21
00
1
kk
kk
kk
kk
x
x




is convergence.
Over interval (-4, 4). Utilizing Gamma-Beta function
relation and binomial coefficients,

1
1
0
(1)(1)
nk
knttd
 
nk
t
Copyright © 2013 SciRes. AJCM
L. P. ZHANG, W. H. JI
32
we calculation sum function of series

 
11
21
000
k
kk
k

11
11
00
00
11
00
00
11
2
00
(22)(1)
=2(1 )+2(1 )
=2(1 )+2(1)'
(1 )
22
1(1)
[1(1 )]
1(1)1
2[1 (1
kkkk
kk kkk k
kk
kk
kk
xxk ttdt
kx ttdtx ttdt
kxtttdt xtttdt
xt ttdt
dt xt t
xt t
xt t
xt




















11
2
00
1
22
0
21(1)
)]
2[1 ]
tdt
tdt xt t
t
tdt
xt xt



Using recursion formulas of integral [7]
1
2
0
44
1
xx
xt xt
1
2
0
22
11 1
2
22 1
44
1
28
arctan ,
4(4) 44
dt
dt
xx
xt xt
x
x
x
xx xx






 
for




04x
2
22
24 4
ln ,
4(4) 44
x
xx
x
x
xxxx x
 

 
for . This completes proof of Theorem 1
Theorem 2 The series of reciprocals of non-central
binomial coefficients
The denoial
co
40x .
1)minator contains 1 odd factor of binom
efficients in the series

21
0
44
=( 1)
(2 3)
m
mm
xD
m
x
x
m

(1)

22
21
0
32 12324
D=(1) 3
(2 5)
m
m
mm
x
x
x
xx
m

 (2)

32
21
0
32
512 644 1
()
3
3
(2 7)
512 644
15
39
m
m
mm
x
x
xx
m
D
x
xx


(3)

432
21
0
432
4096 15369641
()
55
555
(2 9)
4096 512324
35
51525
m
m
mm
x
x
xxx
m
D
x
xxx

 
(4)
2) The denominator contains 2 odd factors of binomial
coefficients in the series

21
0
22
16 816 8
(1)D
(23)(2 5)
3
m
m
mm
x
mm
x
xx
(5)
  

21
0
323 2
(23)(2 7)
128 1611281616
()
31
339
m
m
mm
x
mm
D5
x
xxx x


(6)

21
0
323 2
2564881256 1768
()
31
3
(25)(2 7
39
)
m
m
mm
D5
x
x
xx
x
m
x
m
x
 

(7)

21
0
432
432
2048 2561681
()
15 5
155 5
2048 256162
(23)(2 9
4
35
15 45
)
75
m
m
mm
D
x
xx
x
mm
x
x
xxx
  


(8)

432
4
1
2
2
0
3
1024 3841616 1
()
55
555
1024 12820832
105
5
(25)(2 9)
15 25
m
m
mm
x
D
x
xxx
x
xxx
mm
 

(9)

432
4
1
3
2
2
0
2048 3584 20881
()
515
5155
2048 5126568
1
(27)(2 9)
05
55225
m
m
mm
D
x
xxx
x
xx x
x
mm

 

(10)
3) The denominator contains 3 odd factors of binomial
coefficients in the series

21
0
323 2
64164164804
( )
3
3
D
(23)(2 5)(27
5
39
)
m
m
mm
x
m
x
x
x
mm
x

(11)
xx

 

43
21
0
2
432
512641628 1
()
15 5
15 5 5
512 64304 12
1
(2 3
05
1545 7
)( 2
5
5)(2 9)
m
m
mm
x
mmm
D
x
xxx
x
xxx



(12)
Copyright © 2013 SciRes. AJCM
L. P. ZHANG, W. H. JI
Copyright © 2013 SciRes. AJCM
33

21
0
432
43 2
1024 7044841
()
15 15
15 155
1024 10881
(23)(2 7)(2
76 4
21
15 45
9)
225
m
m
mm
D
x
x
mmm
xxx
x
xx x

 

(13)

432
4
1
3
2
2
0
512 1216 112121
()
515
515
(25)(2 7
5
51270
)(2 9
4 12284
35
5
)
15 225
m
m
mm
D
x
x
mm
x
m
xxx
x
xx



(14)
4) The denominator contains 4 odd factors of binomial
coefficients in the series

432
()
15 15
15 155
256 512
D
x
xxx
   (15)
1
42
2
3
0
256 256 32161
544 16
105
1
(2 3)(2 5)(2 7)(2 9)
545225
m
m
mm
x
mm
x
xx
m
x
m

 

Proof of Theorem
1) Left sides of (0) splitting terms
0
!( 1)!
(2 1)!
n
n
nnx D
n
0n
0
(1
)!!()
!( 1)
(1)
1,
(221)!(211)(1
!
)
n
n
n
nnnnx
n
nn
D
n
x

 
 
put
(2 1)!n
2n
1nm ,
1
0
!( 1)!( 1)(2)
1,
(2 1)!(22)(23)
m
m
mmmm xD
mmm



Multiply both sides by 4
x
, arrive to
0
0(2 1)2 3
m
4!(1)!(231)4
,
(2 1)!(23)
4!(1)! 14
(1) ,
!
m
m
m
mmmxD
xmmx
mm xD
x
mm
x




We obtain (1). Let right sides (1) be.(2). Left
sides of (0) splitting terms
0(2 4 1)!(231)(221)(21)((21)
nnn
nnn
13
(2)!( 1)!( 1)(1)
1
,
n
x
nnnnnnx
D
 
 
put 2nm
2
0
13
!( 1)!( 1)(2)(2)(3)
(21)!(22)(23)(24)(25)
,
m
m
x
mmmmmm x
mmmmm
D
 

2
16
x
, arrive to Multiply both sides by
20
2
20
2
20
2
16 16!(1)!
3(21)!
1116
(1 )(1)
23 25
16 16!(1)!
3(21)!
111 16
(1 )=
2325(23)(25)
16 16!(1)!
3(21)!
3/2 1/216
(1 )
2325
m
m
m
m
m
m
mm x
xm
x
D
mm
x
mm x
xm
x
D
mm mmx
mm x
xm
x
D
mmx

 


  
 

 

Multiply both sides b, arrive to y2
3
2
2
0
3232 23
3
!( 1)!32
(2 1)!(25)
m
m
DD
x
x
mm xD
mm x



We obtain (2). Let ri ght sides ( 2) be .
3) Left sides of (0) splitting terms 5
D
2
0
1310
(3)!( 2)!( 2)(1)(1)(1)
(26 1)!(251)(24 1)(2)
,
1
n
n
xx
nnnnnnnnx
nnn n
D

 
 
3nm
put
3
D
2 3
0
!(1)!(1)( 2)( 3)(2)(3)( 4)
1,
310(21)!(2 2)(23)(2 4)(2 5)(26)(2 7)
m
m
xxmmmmmmmmxD
mmmmmmm
 
 

L. P. ZHANG, W. H. JI
34
2
0
!(1)![(2)(3)](4) m
mmxD

1,
3108(2 1)!(23)(25)(27)
m
xxmm m
mmmm

 
Multiply both sides by 3
64
x
,
32 3
)
7
D
m
x
0
32 3
0
32
646416!(1)!3/ 21/ 2164
[1 ](1
5(21)!23252
3
646416!(1)!3/ 21/ 2164
[1 ]
5(21)!2 32 52 7(2 3)(2 7)(2 5)(2 7)
3
6464 16!(1)!
5(
3
m
m
m
m
m
mm x
xm mm
xx
mm xD
xm mmmmmmm
xx x
mm x
x
xx
 

  


3/ 21/ 2
3
0
15/8 3/43/864
(1) ,
21)! 232527
m
D
mm m
x
 

Multiply both sides by
m
8
3, arrive to
35
32 3
0
512 512 25

6 8!(1)!512
52
15 3(2 1)!(27)
39 3,
m
m
mm x
DD D
xmm
xD
xx
 

we obtain (3).
4) Let right sides of (3) be , Left sides of (0) splitting terms
7
D
23
0
( 4)!(3)!(3)( 2)(1)( 2)(1)(1)
131035(2 81)!(2 71)(2 61)(2)((21)
n
n
xx xnnnnn nnn nn xD
nnn nn

  

Put 4nm
23 4
0
!(1)![(2)(3)(4)](5)
1.
3103516(21)!(23)(2 5)(2 7)(2 9)
m
m
xx xmmmmmmxD
mmmmm

  

4
256
x
, arrive to Multiply both sides by
432 4
0
256256128256!(1)!1111256
(1 )(1)(1)(1),
35(21)!2 32 52 729
35
m
m
mm xD
xm mmmm
xxx x
 

expand to
432 0
256256128256!(1)!11111
[1
35(21)! 23252729(23)(25
35
11111
(2 3)(2 7)(2 3)(2 9)(2 5)(
11
)
5)(7) (23)(25)(
27)(25)(29) 7)(29)
29)
m
m
mm x
xm mmmmm
m
m
xxx
mmmm mm
m
mmm
m
m
   


  

 
(2 3)(2 2mmm

(2
4
1
(2 5)(2 7)(29)
1]
(2 3)(2 5)(27)(2 9)
256
mmm
D
mmmmx



In (0.1), there are 10 fractions containing 2 factors, 10
fractions containing 3 factors, 5 fractions containing 4
and one fraction containing 5 factors.
After the below calculation to (0.1), arrive one series
of reciprocals of non-central binomial coefficients with
the denominators 1, 2, 3, 4, 5 factors
1) All fractions of (0.1) divided into partial fractions,
arrive to
432 0
3/ 43/
m
4
256256128256!(1)!15/83/ 43/8115/8
(1
35(21)! 23252729(23)(29)
35
8256 ,
(2 5)(2 9)(2 7)(29)
m
mm x
xm mmmmmm
xxx
D
mmmm x
  

 
 
Copyright © 2013 SciRes. AJCM
L. P. ZHANG, W. H. JI 35
arrive to
432 4
0
256256128256!(1)!35 /1615/169 /165 /16256
[1 ]
35(21)!2 32 52 72 9
35
m
m
mm xD
xm mmmm
xxx x
 

Multiply both sides by 16 ve to
5arri
35 74432 0
4096 40962048 4096
175
169!( 1)!4096
73
555(21)!(29)
515 25
m
m
mmx
DD DD
mm x
D
x
xxx

 
,
(4), Let right sides (4) be
2) In (0.1), the factions of 2 factos retained, other
fractions divided partial fractions. Then spit term the
fraction containing 2 factions, reserve one during each
s beo partial fraction,
arrive to
We obtain9
D
r
split term, what left haen split int
A) 3
43 53
579
4
2
27
16
23 95256
16 16 16
256256128256
35
35 DD D
Dx
x
xx
DD D
x


B) 37
432 3
57 4
9
256256128256
35
35 25
31
16
15 135
16 16
6
16
DD D
DD
x
xxx
DDx



C) 39
432 3
57 4
9
256256 128256
35
35 25
97
48
15 92
611648x
63
DD D
D
x
xx
DDD
x
 

D) 57
432
256256 128256
35
35
35
DD D
x
xxx 
3
579
4
16
16 1 166x
E)
25671
75 .DDD D
59
4323
579
4
256256 128256
35
35 256
35
16
11 99
16 16 16.
x
xx DDD
DDD
x
D
x



F) 79 3
579
5
15 1 1024
16 16 16
DDD
x
 
432
35
16
13 .
256256 128256
35
3 5DD D
x
xx
D
x

Because D,Dis known, the calculation of
(A) – (F) , we gr, other
3579
,,,DDD
et (5) – (1 0 ),
3) In (0.1), the factions of 3 factos retained
fractions divided partial fractions.
Then spit term the fraction containing 3 factions,
reserve one during each split term, what left has been
split into partial fraction, arrive to
A) 357 3
432
256256 128256
35
36
5
33
1x
xxx DD


57 4
9
19 75
16 16
256
16 .DD x
DD

B) 359 3
5
4
79
32
4
256256 128256
35
35 25
101
48
17913
16
6.
16 48
x
xxx
D
x
DD
DDD

 


D
C) 379 3
5
43
79
2
4
1256 203
48
15 1111
1
56 128256
35
35 2
61648
56 .
DD
DDD
x
xxx
D
x




D
D) 5
432 3
79
4
79
5
256256 128256
35
35 256
35
16
13 133
16 1.
616
DD
DD
x
xxx
DD x

 


D
Because D, is known, the calculation
of (A) – (D), w,
4) In (0.1), the factions of 4 factors retained, other frac-
taining 4 factions,
partial fraction, arrive to
3579
,,,DDDD
e get (11) – (14)
tions divided partial fractions.
Then spit term the fraction con
reserve one during each split term, what left has been
split into
3579
43
74
3
59
2
256256 1282513
6
11
2
6
35
35
256
3.
x
xDD
DD
xx
x
DD



D
Because D, , is known, the calculation
of this expressi.
This completes proof of Theorem 2
3. Some Series of Number Values
In (1) - (15) of theorem 2, put
3579
,,,DDDD
on , we get (15)
1,x

21
0
24 3
32
1
7
m
mm
D

D
Put x1
,

21
0
285
ln ,
525
(1)
m
m
mm
D

51
2
;
we have
Copyright © 2013 SciRes. AJCM
L. P. ZHANG, W. H. JI
36
Corolla ry 1 The series of number values of reciprocal
of non-central binomial coefficients
1)

21
0
1
(2
9
2
43
3)
m
mmm
;
2)

21
0
1
(2 5
28 3
9
)
50;
3
m
mmm

3)

21
0
1148341058;
(2 7945
)
m
mmm

4)

21
0
23483 225158
952
1
(2 9)
m
mmm
5

;
5)

21
0
1
(2 3)(25)
4322
33
m
mmmm

;
6)

21
0
1
(2 3)(2 7)
1078
4345
m
mmmm


;
7)

21
0
203 1634;
345
1
(2 5)(2 7)
m
mmmm


8)

21
09) 15 1575
mm
1
(2 3)(2
1963112054;
mmm

9)

21
0(2 5)(29;
15 17
)5
m
mmmm


1
284318084
10)

21
0
1
(2 7)(29)
1563 267422
m
mmmm

;
51575

11)

21
0
43326;
34
1
5
(2 3)(2 5)(2 7)
m
mmmmm

12)

21
0
1
(2 3)(2 5)(2 9)
44345316;
15 1575
m
mmmmm


13)

21
0
1
(2 3)(2 7)(2 9)
683 38842;
15 1575
m
mmmmm


14)

21
0
1
(2 5)(29)(2 9)
92352306;
15 1575

m
mmmmm

15)

21
1
(2 3)(2
mmm

05)(2 7)(
43 762;
51
2
75
9)
mmmm


Corolla ry 2 Alternating series of number values of re-
ciprocals of non-central binominal coefficients
1)

21
0
(1)
(
85
ln 2;
5
23)
m
m
mmm

2)

21
(5 46
ln ;
5
m
0(25)
mmm
1) 72
3
m
3)

21
0ln
15
(2 7)
m
mmm

(1144 53698;
45
1)m
4)

21
0
183(1)
m
2 5206938
ln ;
5525
(2 9)
m
mmm

5)

21
0
(1)
(2 3)(2 5)
26
85ln ;
3
m
m
mmmm


6)

21
0
(1)
(2 3)(2 7)
56 5902;ln
345
m
m
mmmm
7)

21
0(2 5)(27)ln ;
345
m
mmmm

(1) 136 52194
m
8)

21
0
(1)
(2 3)(2 9)
184 5152294
ln ;
3 1575
m
mmmm


m
9)

21
0
(1)
(2 5)(2 9)
16574
88 5ln;
175
m
m
mmmm


10)

21
0
(1)
(2 7)(29)
664 5375122
ln ;
3 1575
m
m
mmmm


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L. P. ZHANG, W. H. JI
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