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A369390 a(n) = [x^prime(n)] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

%I A369390 #10 Jan 27 2024 20:48:11
%S A369390 1,1,2,4,6,13,31,77,188,449,1191,3014,7920,21498,57833,154073,412733,
%T A369390 1141274,3106771,8576977,24015471,66489615,185886699,517837152,
%U A369390 1435964205,4034697191,11438332340,32395341851,92396549863,263233759500,736127855014,2093027604453
%N A369390 a(n) = [x^prime(n)] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).
%H A369390 Alois P. Heinz, <a href="/A369390/b369390.txt">Table of n, a(n) for n = 1..400</a>
%p A369390 s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
%p A369390 b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
%p A369390       b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
%p A369390     end:
%p A369390 a:= n-> b(ithprime(n), n):
%p A369390 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jan 22 2024
%t A369390 Table[Coefficient[Product[(x^Prime[k] + 1 + 1/x^Prime[k]), {k, 1, n}], x, Prime[n]], {n, 1, 32}]
%Y A369390 Cf. A063890, A316706, A350880.
%K A369390 nonn
%O A369390 1,3
%A A369390 _Ilya Gutkovskiy_, Jan 22 2024