OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Carl McTague, On the Greatest Common Divisor of C(q*n,n), C(q*n,2*n), ...C(q*n,q*n-q), arXiv:1510.06696 [math.CO], 2015.
FORMULA
For prime p>2, valuation(a(n), p) = 1 if 2*n = p^i+p^j for some i<=j, 0 otherwise (see Theorem 2 in McTague).
MATHEMATICA
Table[GCD @@ Array[Binomial[2 n, 2 #] &, {n - 1}], {n, 1, 66}] (* Michael De Vlieger, Dec 09 2015, modified to match the new corrected data by Antti Karttunen, Dec 11 2015 *)
PROG
(PARI) allocatemem(2^30); A265388(n) = if(n<=1, 0, gcd(vector(n-1, k, binomial(2*n, 2*k)))) \\ PARI versions before 2.8 return an erroneous value 1 for gcd of an empty vector/set. - Michel Marcus, Dec 08 2015 and Antti Karttunen, Dec 11 2015
for(n=1, 10000, write("b265388.txt", n, " ", A265388(n)));
(Scheme)
(define (A265388 n) (let loop ((z 0) (k 1)) (cond ((>= k n) z) ((= 1 z) z) (else (loop (gcd z (A007318tr (* 2 n) (* 2 k))) (+ k 1))))))
;; A version using fold. Instead of fold-left we could as well use fold-right:
(define (A265388 n) (fold-left gcd 0 (map (lambda (k) (A007318tr (* 2 n) (* 2 k))) (range1-n (- n 1)))))
(define (range1-n n) (let loop ((n n) (result (list))) (cond ((zero? n) result) (else (loop (- n 1) (cons n result))))))
;; In above code A007318tr(n, k) computes the binomial coefficient C(n, k), i.e., Pascal's triangle A007318. - Antti Karttunen, Dec 11 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Dec 08 2015
STATUS
approved