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A340990 a(n) is the (2n)-th term of the n-fold self-convolution of the primes.

%I A340990 #15 Feb 21 2023 10:48:21
%S A340990 1,3,29,291,3121,34123,379853,4280251,48681569,557686227,6425630909,
%T A340990 74384480019,864461820049,10079577033243,117859582680813,
%U A340990 1381492094548651,16227770995740865,190979248798795427,2251327736286726749,26579050506578504195,314212180691846338801
%N A340990 a(n) is the (2n)-th term of the n-fold self-convolution of the primes.
%H A340990 Alois P. Heinz, <a href="/A340990/b340990.txt">Table of n, a(n) for n = 0..924</a>
%F A340990 a(n) = [x^(2n)] (Sum_{j>=1} prime(j)*x^j)^n.
%F A340990 a(n) = A340991(2n,n).
%p A340990 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, ithprime(n+1),
%p A340990       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A340990     end:
%p A340990 a:= n-> b(n$2):
%p A340990 seq(a(n), n=0..23);
%t A340990 b[n_, k_] := b[n, k] = If[k == 0, 1, If[k == 1, Prime[n + 1], With[{q = Quotient[k, 2]}, Sum[b[j, q] b[n - j, k - q], {j, 0, n}]]]];
%t A340990 a[n_] := b[n, n];
%t A340990 a /@ Range[0, 23] (* _Jean-François Alcover_, Feb 04 2021, after _Alois P. Heinz_ *)
%Y A340990 Cf. A000040, A309955, A340991.
%K A340990 nonn
%O A340990 0,2
%A A340990 _Alois P. Heinz_, Feb 01 2021