Odd numbers in the terms of the central hexanomial coefficients.
1, 1, 27, 24017, 25090131, 27981391815, 32312202610863, 38131413053103057, 45670111932745529235, 55291111957193043094857, 82907103755835453891805670493, 126959848633320332188497187361232663, 158008716644446928751499711727674353081
1
It appears that none of these numbers is prime. The next two sequences show why this sequence is interesting; the central binomial coefficients are even numbers except for those having index 2^k - 1; the central trinomial coefficients are all odd; the central quadrinomial coefficients are even except for the terms having index (4^k - 1)/3 for k = 0, 1, 2,…; and the central pentanomial coefficients are all odd.
T. D. Noe, Plot of 100 terms
T. D. Noe, Table of 100 terms
(Mma) prod = 1; t = Table[prod = Expand[prod*Sum[x^k, {k, 0, 5}]]; Max[CoefficientList[prod, x]], {n, 2^8}]; t = Join[{1}, t]; t2 = Flatten[Position[t, _?(OddQ[#] &)] ] - 1; t[[t2 + 1]]
Cf. A018901, S000978, S000979.
nonn
T. D. Noe, Mar 08 2017