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binomial

binomial(6+5-1,6);

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binomial

binomial(15,5);

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binomial-do-print-sum

S(n,k):=sum(1/(2^(2*n...

for ii:1 thru 4 do (p...

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binomial

binomial(10, 2)^2;

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binomial

binomial(5,-3);

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binomial-gen_laguerre-laguerre-rat-sum

nn:2;

rat(gen_laguerre(nn,a...

sum(binomial(nn,kk)*G...

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binomial

binomial(36,10);

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binomial-genmatrix-invert-makelist-stirling1

h[i,j]:=binomial(i,j)...

M: genmatrix(h,3,3);

invert(M);

Calculate

binomial-ratsimp-subst-sum

a: i;

b: binomial(i,j);

c: x^j;

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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
t1(n) := 2*binomial(n,2)^3*binomial(n-2,2) - binomial(n-2,2)^4;
                         3                                    4
(%o1) t1(n) := 2 binomial (n, 2) binomial(n - 2, 2) - binomial (n - 2, 2)
(%i2) t2(n) := 4*binomial(n,2)^2*binomial(n-2,1)^2;
                                   2               2
(%o2)           t2(n) := 4 binomial (n, 2) binomial (n - 2, 1)
(%i3) t3(n) := 4*binomial(n,2)^2*binomial(n-2,1);
                                    2
(%o3)            t3(n) := 4 binomial (n, 2) binomial(n - 2, 1)
(%i4) t4(n) := binomial(n,2)^2;
                                            2
(%o4)                      t4(n) := binomial (n, 2)
(%i5) t(n) := binomial(n,2)^4;
                                            4
(%o5)                       t(n) := binomial (n, 2)
(%i6) expand(t(n) - t1(n) - t2(n) - t3(n) - t4(n));
           7       6        5                  3         2
          n    29 n    179 n         4   1257 n    1557 n
(%o6)   - -- + ----- - ------ + 153 n  - ------- + ------- - 270 n + 81
          2      4       4                  4         4
(%i7) %plot2d(t(n) - t1(n) - t2(n) - t3(n) - t4(n), [n,2, 10]);
                     4  4                          3  3
              (n - 1)  n    (n - 3) (n - 2) (n - 1)  n                   2  2
(%o7) %plot2d(----------- - --------------------------- - (n - 2) (n - 1)  n
                  16                     8
                                          2  2          4        4
                   2        2  2   (n - 1)  n    (n - 3)  (n - 2)
          - (n - 2)  (n - 1)  n  - ----------- + -----------------, [n, 2, 10])
                                        4               16
(%i8) 
Run Example
binomial(k, i)*binomial(v-k,i)*(i-x-1);
(%o1)           binomial(k, i) binomial(v - k, i) (- x + i - 1)
(%i2) 
[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,ceiling,coeff,expand,rat,sum] [binomial,coeff,do,expand,float,length,limit,makelist,print,product,reverse] [binomial,cos,factor,load,next_prime,prev_prime,sin] [binomial,declare,expand,factorial_expand,gamma,rat,true] [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,float,makelist,print,stirling2,sum] [binomial,do,makelist,print,stirling2,sum] [binomial,do,makelist,sum] [binomial,expand] [binomial,float,sum] [binomial,float] [binomial,floor,rat,sum] [binomial,floor,sum] [binomial,gcd,mod] [binomial,genmatrix,invert] [binomial,linsolve] [binomial,load,makelist] [binomial,load] [binomial,makelist] [binomial,max,sum] [binomial,mod] [binomial,numer,true] [binomial,numer] [binomial,nusum] [binomial,plot2d] [binomial,simpsum,sum,true] [binomial,simpsum,sum] [binomial,simpsum,true] [binomial,solve,sum] [binomial,stirling2,sum] [binomial,sum] [binomial]

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