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A002125 a(n) = Sum_{k=0..n} f(k)*f(n-k) where f(k) = A002124(k).

%I A002125 M0024 N0006 #25 Oct 14 2023 22:36:36
%S A002125 1,0,0,2,0,2,3,2,6,4,9,14,11,26,29,34,62,68,99,140,169,252,322,430,
%T A002125 607,764,1059,1424,1845,2546,3344,4442,6002,7876,10575,14058,18575,
%U A002125 24878,32842,43630,58073,76658,101913,134964,178468,236776,312874,414094,547947,723646
%N A002125 a(n) = Sum_{k=0..n} f(k)*f(n-k) where f(k) = A002124(k).
%C A002125 Arises in studying the Goldbach conjecture.
%D A002125 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002125 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002125 N. J. A. Sloane, <a href="/A002125/b002125.txt">Table of n, a(n) for n = 0..1000</a>
%H A002125 P. A. MacMahon, <a href="http://plms.oxfordjournals.org/content/s2-23/1/290.extract">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [The sequence I_n]
%F A002125 G.f.: 1/(1 - Sum_{k>=2} x^prime(k))^2. - _Ilya Gutkovskiy_, Apr 11 2017
%p A002125 M:=120; f:=array(0..M); f[0]:=1; f[1]:=0; f[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + f[n-p]; fi; od: f[n]:=t1; od: # f is A002124
%p A002125 A002125:=array(0..M); for n from 0 to M do A002125[n]:=add(f[t]*f[n-t],t=0..n); od: [seq(A002125[n],n=0..M)];
%t A002125 CoefficientList[Series[1/(1 - Sum[x^Prime[k], {k, 2, 50}])^2, {x, 0, 50}], x] (* _Indranil Ghosh_, Apr 12 2017 *)
%o A002125 (Haskell)
%o A002125 a002125 n = a002125_list !! n
%o A002125 a002125_list = uncurry conv $ splitAt 1 a002124_list where
%o A002125    conv xs (z:zs) = sum (zipWith (*) xs $ reverse xs) : conv (z:xs) zs
%o A002125 -- _Reinhard Zumkeller_, Mar 21 2014
%K A002125 nonn
%O A002125 0,4
%A A002125 _N. J. A. Sloane_
%E A002125 Edited by _N. J. A. Sloane_, Dec 03 2006