Tue Feb 18 20:35:25 MST 1997 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Combinatorial Theory ---------- *) 0. Second In[6]:= (* Pochhammer symbol (a)_n = a (a + 1) ... (a + n - 1) => a (a + 1)\ > (a + 2) *) 0. Second In[7]:= Pochhammer[a, 3] 0. Second Out[7]= a (1 + a) (2 + a) In[8]:= (* Binomial coefficient => n (n - 1) (n - 2)/6 *) 0. Second In[9]:= Binomial[n, 3] 0. Second (-2 + n) (-1 + n) n Out[9]= ------------------- 6 In[10]:= (* 2^n n! (2 n - 1)!! => (2 n)! *) 0. Second In[11]:= 2^n * n! * (2*n - 1)!! 0.01 Second n Out[11]= 2 n! (-1 + 2 n)!! In[12]:= (* 2^n n! product(2 k - 1, k = 1..n) => (2 n)! *) 0. Second In[13]:= 2^n * n! * Product[2*k - 1, {k, 1, n}] 0.21 Second 2 n 1 2 n! Gamma[- + n] 2 Out[13]= -------------------- Sqrt[Pi] In[14]:= FullSimplify[%] 0.38 Second Out[14]= Gamma[1 + 2 n] In[15]:= (* => (2 n)!/[2^(2 n) (n!)^2] or (2 n - 1)!!/[2^n n!] *) 0. Second In[16]:= Gamma[n + 1/2]/(Sqrt[Pi] * n!) 0. Second 1 Gamma[- + n] 2 Out[16]= ------------ Sqrt[Pi] n! In[17]:= (* Partitions of an integer => {1+1+1+1, 1+1+2, 1+3, 2+2, 4} (5 in\ > all) *) 0. Second In[18]:= PartitionsP[4] 0. Second Out[18]= 5 In[19]:= (* Stirling numbers of the first kind: S_1(5, 2) => -50 *) 0. Second In[20]:= StirlingS1[5, 2] 0. Second Out[20]= -50 In[21]:= (* Euler's totient function => 576 *) 0. Second In[22]:= EulerPhi[1776] 0.01 Second Out[22]= 576 In[23]:= (* ---------- Quit ---------- *) 0. Second In[24]:= Quit[] real 2.04 user 1.31 sys 0.38