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%I #5 Dec 05 2013 19:56:46
%S 1,3,2,5,14,42,6,429,1430,4862,16796,58786,2261,742900,2674440,66861,
%T 35357670,129644790,2274470,1767263190,25246617,24466267020,
%U 91482563640,343059613650,1289904147324,4861946401452,18367353072152,69533550916004
%N a(1) = 1; a(n) = floor {(n+1)(n+2)(n+3)...(n+k)}/{(n-1)(n-2)(n-3)...(n-k)} for the least value of k.
%C The first occurrence of an integer in the sequence (n+1)/(n-1), (n+1)(n+2)/{(n-1)(n-2)}, (n+1)(n+2)(n+3)/{(n-1)(n-2)(n-3)},... The triangle of these numbers with initial value of n = 2, for k = 1 to n-1 is:
%C 3
%C 2 10
%C 5/3 5 35
%C 3/2 7/2 14 126
%C 7/5 14/5 42/5 42 462
%C 4/3 12/5 6 22 132 1716
%C ...
%C Sequence contains the first integer in each row.
%C The leading diagonal of the triangle is given by A001700 = C(2n+1,n+1). i.e. eventually an integer occurs for k < n-1.
%e a(6)= 42: the relevant numbers are 7/5, 7*8/(5*4), 7*8*9/(5*4*3), (7*8*9*10)/(5*4*3*2),...or 1.4, 2.8,8.4,42,...
%o (PARI) { a(n) = local(p, q, r); p=1; q=1; for(k=1,n, p*=n+k; q*=n-k; r=gcd(p,q); p\=r; q\=r; if(q==1,break)); return(p) } (Alekseyev)
%Y Cf. A001700.
%Y Sequence of corresponding values of k is A103634
%K nonn
%O 1,2
%A _Amarnath Murthy_, Mar 22 2004
%E More terms from _Max Alekseyev_, Feb 11 2005
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