Numbers n such that 5 consecutive terms of binomial(n,k) satisfy a cubic polynomial.
13, 62, 1517, 15039
1
Is this sequence finite? There are no additional terms less than 1000000.
T. D. Noe, Plot of 4 terms
Example: for the case n = 13, the cubic polynomial (-143/6)*(2*x^3 - 27*x^2 - 97*x - 114) computes the binomial coefficients for x = 4..8.
(Mma) t = {}; Do[d = Differences[Binomial[n, Range[0, n]], 4]; If[MemberQ[d, 0], AppendTo[t, n]], {n, 1600}]; t
Cf. A241199 (quadratic).
nonn,hard,more
T. D. Noe, Apr 21 2016