This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007443 M1436 #56 Oct 31 2023 17:54:28
%S A007443 2,5,13,33,83,205,495,1169,2707,6169,13889,30993,68701,151469,332349,
%T A007443 725837,1577751,3413221,7349029,15751187,33616925,71475193,151466705,
%U A007443 320072415,674721797,1419327223,2979993519,6245693407,13068049163
%N A007443 Binomial transform of primes.
%C A007443 Equals row sums of triangle A164738. Example: a(4) = 33 = sum of terms in row 4 of triangle A164738: (2, 3, 5, 3, 5, 7, 5, 3). - _Gary W. Adamson_, Aug 23 2009
%C A007443 It might have been more natural to define this sequence with offset 0, which would also make the formula simpler. Then a(n) would be the first term of the sequence obtained from the primes by applying n times the operation "take sums of successive terms", Ts(k) = s(k)+s(k+1). - _M. F. Hasler_, Jun 02 2017
%D A007443 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007443 Seiichi Manyama, <a href="/A007443/b007443.txt">Table of n, a(n) for n = 1..3000</a> (terms 1..1000 from Vincenzo Librandi)
%H A007443 Vaclav Kotesovec, <a href="/A007443/a007443.jpg">Plot of a(n) / (2^n * n * log(n)) for n = 2..20000</a>
%H A007443 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A007443 a(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). - _M. F. Hasler_, Jun 02 2017
%F A007443 G.f.: Sum_{k>=1} prime(k)*x^k/(1 - x)^k. - _Ilya Gutkovskiy_, Apr 21 2019
%p A007443 a:=n->add(binomial(n-1,k-1)*ithprime(k),k=1..n): seq(a(n),n=1..30); # _Muniru A Asiru_, Oct 23 2018
%t A007443 A007443[n_]:=Sum[Binomial[n-1,k-1]Prime[k],{k,n}];Array[A007443,50] (* or *)
%t A007443 Module[{nmax=50,b},b=Prime[Range[nmax]];Join[{2},Table[First[b=ListConvolve[{1,1},b]],nmax-1]]] (* _Paolo Xausa_, Oct 31 2023 *)
%o A007443 (PARI) A007443(n)=sum(k=1,n,binomial(n-1,k-1)*prime(k)) \\ _M. F. Hasler_, Jun 02 2017
%Y A007443 Cf. A164738.
%Y A007443 Cf. A001043, A096277, A096278, A096279. See A287915 for indices of primes.
%Y A007443 First differences give A178167.
%K A007443 nonn,easy
%O A007443 1,1
%A A007443 _N. J. A. Sloane_
%E A007443 More terms from _Vladimir Joseph Stephan Orlovsky_, May 21 2010