Wed Apr 16 20:31:35 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Combinatorial Theory ---------- -- Pochhammer symbol (a)_n = a (a + 1) ... (a + n - 1) => a (a + 1) (a + 2) --pochhammer(a, 3) -- Binomial coefficient => n (n - 1) (n - 2)/6 binomial(n, 3) n (1) ( ) 3 Type: Expression Integer Time: 0.27 (IN) + 0.12 (EV) + 0.20 (OT) + 0.12 (GC) = 0.70 sec factorFraction(normalize(%) :: Fraction Polynomial Integer) (n - 2)(n - 1)n (2) --------------- 6 Type: Fraction Factored Polynomial Integer Time: 0.12 (IN) + 0.38 (EV) + 0.25 (OT) + 0.07 (GC) = 0.82 sec -- 2^n n! (2 n - 1)!! => (2 n)! --2**n * factorial(n) * (2*n - 1)!! -- 2^n n! product(2 k - 1, k = 1..n) => (2 n)! 2**n * factorial(n) * product(2*k - 1, k = 1..n) n n++-++ (3) n!2 | | 2k - 1 | | k= 1 Type: Expression Integer Time: 0.30 (IN) + 0.03 (EV) + 0.07 (OT) = 0.40 sec -- => (2 n)!/[2^(2 n) (n!)^2] or (2 n - 1)!!/[2^n n!] Gamma(n + 1/2)/(sqrt(%pi) * factorial(n)) _ 2n + 1 | (------) 2 (4) ---------- +---+ n!\|%pi Type: Expression Integer Time: 0.85 (IN) + 0.02 (EV) + 0.07 (OT) + 0.02 (GC) = 0.95 sec -- Partitions of an integer => {1+1+1+1, 1+1+2, 1+3, 2+2, 4} (5 in all) 4 (5) 4 Type: PositiveInteger Time: 0 sec -- Stirling numbers of the first kind: S_1(5, 2) => -50 stirling1(5, 2) (6) - 50 Type: Integer Time: 0.02 (EV) = 0.02 sec -- Euler's totient function => 576 eulerPhi(1776) (7) 576 Type: PositiveInteger Time: 0.02 (IN) = 0.02 sec -- ---------- Quit ---------- )quit real 9.9 user 3.6 sys 0.3