%I #24 Dec 04 2018 07:43:23
%S 1,2,1,12,9,2,60,54,20,3,840,840,400,105,12,2520,2700,1500,525,108,10,
%T 27720,31185,19250,8085,2268,385,30,360360,420420,280280,133770,45864,
%U 10780,1560,105,720720,864864,611520,321048,127008,36960,7488,945,56,12252240,15036840,11138400,6297480,2776032,942480
%N Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^i*c(n,i)*binomial(i*p,p) is divisible by p^(2n-1).
%H R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:<a href="http://doi.org/10.1007/s10958-018-3948-0">10.1007/s10958-018-3948-0</a>; also <a href="http://arxiv.org/abs/1602.02632">arXiv</a>, arXiv:1602.02632 [math.NT], 2016-2018.
%F c(n,i) = A003418(2*(n-1))*binomial(2*n-1,n-i)*(2*i-1)/i/binomial(2*n-1,n).
%e n=1: 1
%e n=2: 2, 1
%e n=3: 12, 9, 2
%e n=4: 60, 54, 20, 3
%e n=5: 840, 840, 400, 105, 12
%e ...
%e For all primes p>3, p^3 divides 2 - binomial(2*p,p) (cf. A087754).
%e For all primes p>5, p^5 divides 12 - 9*binomial(2*p,p) + 2*binomial(3*p,p) (cf. A268589).
%e For all primes p>7, p^7 divides 60 - 54*binomial(2*p,p) + 20*binomial(3*p,p) - 3*binomial(4*p,p) (cf. A268590).
%t a3418[n_] := LCM @@ Range[n];
%t c[1, 1] = 1; c[n_, i_] := a3418[2(n-1)] Binomial[2n-1, n-i] ((2i-1)/i/ Binomial[2n-1, n]);
%t Table[c[n, i], {n, 1, 10}, {i, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 04 2018 *)
%o (PARI) { A268512(n,i) = lcm(vector(2*(n-1),i,i)) * binomial(2*n-1,n-i) * (2*i-1) / i / binomial(2*n-1,n) }
%Y Cf. A099996 (first column), A068550 (diagonal), A087754, A268589, A268590, A254593.
%K nonn,tabl
%O 1,2
%A _Max Alekseyev_, Feb 06 2016