### Related

##### binomial-expand-sum

expand((1-x)^3-(1+x)^...

expand((1-(1-2*x)^3)/2);

sum(binomial(3,2*i-1)...

Calculate

##### binomial-numer

1-(35/36)^24, numer;

1-binomial(24,0)*(1/3...

1-binomial(24,24)*(1/...

Calculate

binomial(10,3);

120/10^3,numer;

Calculate

##### binomial-gcd-mod

a: mod(binomial(41040...

b: gcd(41041,a);

c: mod(binomial(1000,...

Calculate

Serie:random_binomial...

Calculate

float(1-cdf_binomial(...

pdf_binomial (2,3,1/6);

Calculate

##### binomial-floor-rat-sum

N:2;

substlist: [w=(1+z)/2];

z1: rat(sum(a[2*n]/2^...

Calculate

##### binomial

binomial(1/2, 3);

Calculate

##### binomial-expand-factorial_expand-subst-true

factorial_expand: true;

f: binomial(b+i,b)*x^...

subst([i=i+1],f)/f;

Calculate

sin(%pi/2);

cos(%pi/4);

binomial(7,3);

Calculate

### binomial

Run Example
```(%i1)declare(m,integer);
(%o1)                                done
(%i2) factorial_expand:true;
(%o2)                                true
(%i3) epsilon(n,k):=(binomial(n,k)*gamma(3/2)/gamma(3/2-n+k)-(-1)^n*binomial(n-1,k-1)*n!/k!);
3
binomial(n, k) gamma(-)
2
(%o3) epsilon(n, k) := -----------------------
3
gamma(- - n + k)
2
n
(- 1)  binomial(n - 1, k - 1) n!
- --------------------------------
k!
(%i4) G(m):=-m*(4*m-3)/2;
(- m) (4 m - 3)
(%o4)                       G(m) := ---------------
2
(%i5) rat(epsilon(2*m,2*m-4)-G(m)*epsilon(2*m-1,2*m-4));
8         7          6          5          4          3
(%o5)/R/ (1024 m  - 8704 m  + 30976 m  - 59680 m  + 67036 m  - 43756 m
2
+ 15309 m  - 2205 m)/96
(%i6) epsilon(4,1)-G(2)*epsilon(3,1);
15
(%o6)                                 --
4
(%i7) epsilon(2,1);
(%o7)                                 - 1
(%i8) epsilon(2,0);
1
(%o8)                                 - -
4
(%i9) ```
Run Example
```binomial(k, i)*binomial(v-k,i)*(i-x-1);
(%o1)           binomial(k, i) binomial(v - k, i) (- x + i - 1)
(%i2) ```
Run Example
```remvalue(n,z);
(%o1)                           [false, false]
(%i2) matchdeclare(a,lambda([x],freeof(x,n)));
(%o2)                                done
(%i3) matchdeclare(b,lambda([x],freeof(x,n) and integerp(x) and x>
0));
(%o3)                                done
(%i4) matchdeclare(f,all);
(%o4)                                done
(%i5) matchdeclare(g,lambda([x],freeof(n-1,x)));
(%o5)                                done
(%i6) defrule(r1,zt(1,n,z),z/(z-1));
z
(%o6)                      r1 : zt(1, n, z) -> -----
z - 1
(%i7) defrule(r2,zt(n,n,z),z/(z-1)^2);
z
(%o7)                    r2 : zt(n, n, z) -> --------
2
(z - 1)
(%i8) defrule(r3,zt(1/n,n,z),log(z/(z-1)));
1                 z
(%o8)                   r3 : zt(-, n, z) -> log(-----)
n               z - 1
(%i9) defrule(r4,zt(a^n,n,z),z/(z-a));
n             z
(%o9)                     r4 : zt(a , n, z) -> -----
z - a
(%i10) defrule(r5,zt(a^(n-1),n,z),1/(z-a));
n - 1             1
(%o10)                  r5 : zt(a     , n, z) -> -----
z - a
(%i11) defrule(r6,zt(binomial(n,b),n,z),z/(z-1)^(b+1));
- b - 1
(%o11)         r6 : zt(binomial(n, b), n, z) -> (z - 1)        z
(%i12) defrule(r7,zt(n^b,n,z),block([tmp:z/(z-1)],  for i:1 thru b do tmp:-z*diff(tmp,z),  factor(tmp)));
b                          z
(%o12) r7 : zt(n , n, z) -> block([tmp : -----],
z - 1
for i thru b do tmp : - diff(tmp, z) z, factor(tmp))
(%i13) defrule(r8,zt(sin(a*n),n,z),z*'sin(a)/((z^2-2*z*'cos(a)+1)));
sin(a) z
(%o13)          r8 : zt(sin(a n), n, z) -> -------------------
2
z  - 2 cos(a) z + 1
(%i14) defrule(r9,zt(cos(a*n),n,z),z*(z-'cos(a))/(z^2-2*z*'cos(a)+1));
z (z - cos(a))
(%o14)          r9 : zt(cos(a n), n, z) -> -------------------
2
z  - 2 cos(a) z + 1
(%i15) defrule(r10,zt(kron_delta(b,n),n,z),1/z^b);
1
(%o15)              r10 : zt(kron_delta(b, n), n, z) -> --
b
z
(%i16) defrule(r11,zt(a^n*f,n,z),'subst(z/a,z,ztrans(f,n,z)));
n                        z
(%o16)     r11 : zt(a  f, n, z) -> substitute(-, z, ztrans(f, n, z))
a
(%i17) defrule(r12,zt(n^b*g,n,z),block([tmp:ztrans(g,n,z)],  for i:1 thru b do tmp:-z*'diff(tmp,z),  return(tmp)));
b
(%o17) r12 : zt(g n , n, z) -> block([tmp : ztrans(g, n, z)],
dtmp
for i thru b do tmp : - ---- z, return(tmp))
dz
(%i18) defrule(r13,zt(exp(-a*n),n,z),z/(z-exp(-a)));
- a n               z
(%o18)               r13 : zt(%e     , n, z) -> ---------
- a
z - %e
(%i19) defrule(r14,zt(f(n-m),n,z),z^(-m)*ztrans(f(n),n,z));
ztrans(f(n), n, z)
(%o19)          r14 : zt(f(n - m), n, z) -> ------------------
m
z
(%i20) declare(zt,linear);
(%o20)                               done
(%i21) ztrans1(e):=apply1(e,r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12,r13,r14);
(%o21) ztrans1(e) := apply1(e, r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11,
r12, r13, r14)
(%i22) ztrans(f,n,z):=ev(ztrans1(zt(f,n,z)),diff,subst);
(%o22)     ztrans(f, n, z) := ev(ztrans1(zt(f, n, z)), diff, subst)
(%i23) defrule(ir1,izt(z/(z-a),z,n),a^n);
z              n
(%o23)                   ir1 : izt(-----, z, n) -> a
z - a
(%i24) defrule(ir2,izt(1/(z-a),z,n),a^(n-1));
1              n - 1
(%o24)                 ir2 : izt(-----, z, n) -> a
z - a
(%i25) declare(izt,linear);
(%o25)                               done
(%i26) invztrans1(e):=apply1(e,ir1,ir2);
(%o26)               invztrans1(e) := apply1(e, ir1, ir2)
(%i27) invztrans(f,z,n):=invztrans1(izt(f,z,n));
(%o27)          invztrans(f, z, n) := invztrans1(izt(f, z, n))
(%i28) ztrans(1,n,z);
z
(%o28)                               -----
z - 1
(%i29) ztrans(a,n,z);
a z
(%o29)                               -----
z - 1
(%i30) expand(ztrans((-1)^n,n,z));
z
(%o30)                               -----
z + 1
(%i31) ztrans(n,n,z);
z
(%o31)                             --------
2
(z - 1)
(%i32) ztrans(n^2,n,z);
z (z + 1)
(%o32)                             ---------
3
(z - 1)
(%i33) ztrans(a^(n-1),n,z);
1
(%o33)                               -----
z - a
(%i34) factor(ztrans(a^(n-1)+n^2,n,z));
3      2      2
2 z  - a z  - 2 z  - a z + 3 z - 1
(%o34)                ----------------------------------
3
(z - 1)  (z - a)
(%i35) ```

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