A007443 - OEIS

%I M1436 #56 Oct 31 2023 17:54:28

%S 2,5,13,33,83,205,495,1169,2707,6169,13889,30993,68701,151469,332349,

%T 725837,1577751,3413221,7349029,15751187,33616925,71475193,151466705,

%U 320072415,674721797,1419327223,2979993519,6245693407,13068049163

%N Binomial transform of primes.

%C 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 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A007443/b007443.txt">Table of n, a(n) for n = 1..3000</a> (terms 1..1000 from Vincenzo Librandi)

%H Vaclav Kotesovec, <a href="/A007443/a007443.jpg">Plot of a(n) / (2^n * n * log(n)) for n = 2..20000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). - _M. F. Hasler_, Jun 02 2017

%F G.f.: Sum_{k>=1} prime(k)*x^k/(1 - x)^k. - _Ilya Gutkovskiy_, Apr 21 2019

%p 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[n_]:=Sum[Binomial[n-1,k-1]Prime[k],{k,n}];Array[A007443,50] (* or *)

%t 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 (PARI) A007443(n)=sum(k=1,n,binomial(n-1,k-1)*prime(k)) \\ _M. F. Hasler_, Jun 02 2017

%Y Cf. A164738.

%Y Cf. A001043, A096277, A096278, A096279. See A287915 for indices of primes.

%Y First differences give A178167.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Vladimir Joseph Stephan Orlovsky_, May 21 2010