﻿ Binomial, factorial, double factorial Binomial factorial, double factorial
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Freeman 2009-06-19-10-46

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```
<a name="docA001">
2009-09-22-18-31 start
This file http://freeman2.com/binomial.htm
is part of http://freeman2.com/tute0010.htm
Because binomial, factorial, double factorial
are frequently used functions. Create a file
for them.

Thank you for visiting Freeman's web site.
Freeman (Liu, Hsinhan) 2009-09-22-18-35
```

<a name="factorial">
Output may contain error, Please verify first.
Program environment is MSIE 6.0, please use MSIE

Find binomial, factorial, double factorial, summation factorial. 9808122120
2009-04-27-17-04 binomial(m,n)=m!/[n!*(m-n)!] ; m≧n ; 0!=1
binomial m , n = answer is here

m!=m!!*(m-1)!! ex. 7!=7!!*6!!=7*6*5*4*3*2*1
double factorial: odd 7!!=1*3*5*7 ； even 6!!=2*4*6
sum. fact. for 7=1+2+3+4+5+6+7=28
Please input a number in m box
Gamma function Γ(x)= =(x-1)!
<a name="binomialSq">
Use program to verify page 229 line 5 equation 9808071733 record
Sum k from k=0 to k=n. After whole range summation, answer has no k factor.

 k=n ∑ k=0 ( n 　 k ) 2 ＝ k=n ∑ k=0 ( n 　 k ) ( n 　 n－k ) ＝ ( 2n 　 n )
---page 229 line 5
width of above equation
Above two boxes control equation space.
Next two boxes are program input parameters.
binomial n power = Please see page 229 line 5 eqn.
Someone proved that power≧3 no closed form sol. .
 Box 1 output Box 2 debug
```<a name="ch01c034">
2009-08-08-16-34 start
2009-08-07-22-46 access
http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8612.pdf
Factors of sums of powers of binomial coefficients
by Neil J. Calkin (Clemson, S.C.)
save as binoSum0.pdf This file page 1
(total 10 pages) say
[[
It is possible to show (Wilf, personal
communication, using techniques in [8])
that for 3≦ power ≦9 there is no closed
form for fn,b as a sum of a fixed number
of hypergeometric terms.
]]
2009-08-19-21-01 LiuHH note:
[[
In Neil J. Calkin paper, fn,b was written
as fn,a . LiuHH change 'a' to 'b' avoid
confuse with two following 'a'.
Please see page 229 line 5 left side
expression. Change power '2' to 'a'
then it is same as fn,a 2009-08-19-21-11
]]

Please see page 229 line 5 eqn. Power 2
change to 3 then no closed form solution,
only numerical solution available.
2009-08-08-16-40 stop
```

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Binomial factorial, double factorial
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http://freeman2.com/binomial.htm

Program name in Chinese: (2009-09-20-18-02)
Brief name: 二項數係數
Full name: 二項數係數

Thank you for visiting Freeman's page.
Freeman 　2009-09-22-18-36

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