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%I #16 Oct 19 2018 15:49:13
%S 1,4,10,22,43,80,137,222,343,508,737,1030,1411,1888,2477,3198,4059,
%T 5096,6297,7702,9327,11176,13301,15682,18355,21344,24673,28358,32411,
%U 36896,41769,47082,52883,59148,65937,73298,81251,89776,98957
%N Self-convolution of (1, p(1), p(2), ...).
%C p(1),p(2),p(3)... are the prime numbers (A000040). The analogous sequence for the partition numbers is A048574.
%H Reinhard Zumkeller, <a href="/A023626/b023626.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f: x*(1+b(x))^2 = (c(x)^2)/x, where b(x) and c(x) are respectively the g.f. of A000040 and A008578. - _Mario C. Enriquez_, Dec 10 2016
%e G.f. = x + 4*x^2 + 10*x^3 + 22*x^4 + 43*x^5 + 80*x^6 + 137*x^7 + ...
%t z = 100; p = Join[{1}, Prime[Range[z]]];
%t a[n_] := Sum[p[[i]] p[[n - i + 1]], {i, 1, n}];
%t Table[a[n], {n, 1, z}] (* _Clark Kimberling_, Dec 01 2016 *)
%t a[ n_] := If[ n < 1, 0, SeriesCoefficient[ (1 + O[x]^n + Sum[ Prime[k] x^k, {k, n - 1}])^2, {x, 0, n - 1}]]; (* _Michael Somos_, Dec 01 2016 *)
%t Table[With[{c=Join[{1},Prime[Range[n]]]},ListConvolve[c,c]],{n,0,40}]// Flatten (* _Harvey P. Dale_, Oct 19 2018 *)
%o (Haskell)
%o a023626 n = a023626_list !! (n-2)
%o a023626_list = f a000040_list [1] where
%o f (p:ps) rs = (sum $ zipWith (*) rs a008578_list) : f ps (p : rs)
%o -- _Reinhard Zumkeller_, Nov 09 2015
%Y Cf. A000040, A008578, A014342, A048574.
%K nonn
%O 1,2
%A _Clark Kimberling_
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