Sequence A007443, the binomial transform of the primes A000040, is defined as A007443(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). This is also the first column of the infinite square array T(m,n) with T(1,n) = prime(n) and T(m+1,n) = T(m,n) + T(m,n+1), as for binomial coefficients. Successive rows result from applying this operation of taking the sum of successive terms. So it would be more natural to use index 0 for the first term of this sequence (which is also the only even term, and results from applying the operation 0 times to the primes). This would yield the sequence 0, 1, 2, 4, 8, 21, 24, 27, 31, 40, 98, 103, 137, 182, ...
The next term, if it exists, is greater than 20000. - Vaclav Kotesovec, Dec 19 2020
Any subsequent terms are > 10^5. - Lucas A. Brown, Mar 18 2024
