Numbers n such that w(n) = sum_{k=0..n} binomial(n,k) binomial(n+k,k)/(2*k-1) is prime.
2, 9, 11, 42, 44, 179, 251
1
The function w(n) is tabulated in A245769. The prime value of w(n) for these n are given in S000212.
T. D. Noe, Plot 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[Range[0, 500], PrimeQ[w[#]] &]
nonn
T. D. Noe, Aug 25 2014