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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This page, binomial, was part of Inequality file four.
http://freeman2.com/binomial.htm
First upload 2009-09-23

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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