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binomial-num-rat-subst-sum

N: 2;

xx: num(rat(sum(a[2*i...

yy: num(rat(sum(a[2*i...

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binomial-nusum

nusum ( binomial(n,h)...

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binomial

binomial(3, 1);

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binomial-solve-sum

eq1:sum(binomial(5,i)...

solve(eq1);

Calculate

binomial-simpsum

? simpsum;

sum(binomial(2*h,h+i)...

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binomial-distrib-float-load

load (distrib);

float(1-cdf_binomial(...

pdf_binomial (2,3,1/6);

Calculate

binomial-rat-sum

N:2;

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

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

Calculate

binomial

Run Example
(%i1)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) 
Run Example
n:3;
(%o1)                                  3
(%i2) p:1/2;
                                       1
(%o2)                                  -
                                       2
(%i3) W(k):=binomial(n,k)*p^k*(1-p)^(n-k);
                                            k        n - k
(%o3)               W(k) := binomial(n, k) p  (1 - p)
(%i4) P:sum(W(k),k,0,3);
(%o4)                                  1
(%i5) 
[apply1,binomial,block,cos,defrule,delta,diff,ev,exp,factor,freeof,kron_delta,lambda,linear,log,matchdeclare,remvalue,return,sin,subst] [assume,binomial,simpsum] [bfloat,binomial,block,distrib,floor,load,sum] [bfloat,binomial,sum] [binomial,block,ceiling,diff,expand,ratsubst,return,sum] [binomial,block,delta,determinant,do,expand,genmatrix,hipow,kron_delta,makelist,part,print,return,subst] [binomial,block,delta,determinant,do,expand,genmatrix,kron_delta,return] [binomial,block,delta,determinant,expand,genmatrix,hipow,kron_delta,listofvars,makelist,part,sum] [binomial,block,delta,determinant,expand,genmatrix,kron_delta,lambda,makelist,subst,sum] [binomial,block,genmatrix,return] [binomial,buildq,delta,kron_delta,lambda] [binomial,ceiling,coeff,expand,rat,sum] [binomial,coeff,do,if,inf,makelist,plot2d,printf,sum,taylor] [binomial,cos,factor,load,next_prime,prev_prime,simpsum,sin,sqrt,sum,true] [binomial,cos,factor,load,next_prime,prev_prime,sin] [binomial,delta,genmatrix,kron_delta] [binomial,distrib,find_root,float,load] [binomial,distrib,float,load,lsum,print] [binomial,distrib,float,load,lsum] [binomial,distrib,float,load] [binomial,distrib,floor,load,numer,sqrt,sum] [binomial,distrib,load,makelist] [binomial,distrib,load] [binomial,do,expand,makelist,sum] [binomial,do,makelist,print,rhs,solve,subst,sum] [binomial,do,makelist,sum] [binomial,expand] [binomial,factorial,float,sum] [binomial,float,sqrt] [binomial,float,sum] [binomial,float] [binomial,floor,rat,sum] [binomial,gcd,mod] [binomial,genmatrix,invert] [binomial,linsolve] [binomial,load,makelist] [binomial,load] [binomial,makelist,plot2d] [binomial,mod] [binomial,numer,true] [binomial,numer] [binomial,nusum] [binomial,simpsum,sum,true] [binomial,simpsum,sum] [binomial,simpsum,true] [binomial,simpsum] [binomial,solve,sum] [binomial,subst,sum] [binomial,sum] [binomial]

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