Primes w(n) for the n in S000211.
7, 132689, 3242377, 216814389336723410853887753623, 6861154742528953501789871492999
1
T. D. Noe, Plot of 7 terms
T. D. Noe, Table of 7 terms
Zhi-Wei Sun, A new kind of numbers and their arithmetic properties, arXiv:1408.5381 (Aug 21 2014).
(Mma) w[n_] := Sum[Binomial[n, k] Binomial[n + k, k]/(2 k - 1), {k, 0, n}]; Select[Table[w[n], {n, 0, 500}], PrimeQ]
nonn
T. D. Noe, Aug 25 2014