<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