cp's OEIS Frontend

(-1)^n*a(n) > 0, and Sum_{k=0..n} C(n,k)*C(n+k,k)*C(2k,k)*C(2n-2k,n-k)*(-1)^(n-k) = Sum_{k=0..n}C(n,k)^4.

Conjecture 1: Sum_{k>=0}(4k+1) a(k)/(-48)^k = sqrt(72+42*sqrt(3))/Pi.

Conjecture 2: For each n > 0, the number (Sum_{k=0..n-1} (-1)^k*(4k+1)*48^(n-1-k)*a(k))/n is a positive integer.

Conjecture 3: For any prime p > 3, the square of (Sum_{k=0..p-1} (4k+1)a(k)/(-48)^k)/p is congruent to 14*(3/p)-(p/3)-12 modulo p, where (a/p) is the Legendre symbol.

Conjecture 4: Let p > 3 be a prime, and let S(p) = Sum_{k=0..p-1} a(k)/(-48)^k. If p == 1 (mod 4) and p = x^2 + 4y^2 with x and y integers, then S(p) == 4x^2-2p (mod p^2). If p == 3 (mod 4), then S(p) == 0 (mod p^2).