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A014345 Exponential convolution of primes with themselves.

%I A014345 #21 Sep 08 2022 08:44:39
%S A014345 4,12,38,118,362,1082,3166,8910,24426,64226,165262,413418,1021362,
%T A014345 2490686,6009150,14401410,34098042,80281962,187356750,432549154,
%U A014345 992941250,2256712462,5088826238,11408805862,25425739346,56383362854,124565557898,274390550594
%N A014345 Exponential convolution of primes with themselves.
%H A014345 Alois P. Heinz, <a href="/A014345/b014345.txt">Table of n, a(n) for n = 0..1000</a>
%F A014345 E.g.f.: (Sum_{k>=0} prime(k+1)*x^k/k!)^2. - _Ilya Gutkovskiy_, Mar 10 2018
%F A014345 a(n) = Sum_{j=0..n} binomial(n,j)*prime(j+1)*prime(n-j+1). - _G. C. Greubel_, Jun 07 2019
%p A014345 a:= proc(n) option remember; (p-> add(
%p A014345       p(j+1)*p(n-j+1)*binomial(n, j), j=0..n))(ithprime)
%p A014345     end:
%p A014345 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 10 2018
%t A014345 a[n_] := Sum[Prime[j + 1] Prime[n - j + 1] Binomial[n, j], {j, 0, n}];
%t A014345 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 05 2018, from Maple *)
%o A014345 (Magma) [&+[NthPrime(k+1)*NthPrime(n-k+1)*Binomial(n, k): k in [0..n]]: n in [0..30]]; // _Vincenzo Librandi_, Jun 07 2019
%o A014345 (PARI) {a(n) = sum(j=0,n, binomial(n,j)*prime(j+1)*prime(n-j+1))}; \\ _G. C. Greubel_, Jun 07 2019
%o A014345 (Sage) [sum(binomial(n,j)*nth_prime(j+1)*nth_prime(n-j+1) for j in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 07 2019
%Y A014345 Cf. A000040, A014347, A014352.
%K A014345 nonn
%O A014345 0,1
%A A014345 _N. J. A. Sloane_