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A369560 a(n) = [x^n] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

%I A369560 #6 Jan 25 2024 19:57:14
%S A369560 1,0,1,2,3,6,16,38,91,225,547,1407,3570,9250,24578,65740,175626,
%T A369560 470084,1279101,3482419,9547953,26445796,73251187,203818706,567543095,
%U A369560 1577629707,4408095456,12400615844,34995570604,99241500366,282037360250,795846583187
%N A369560 a(n) = [x^n] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).
%C A369560 a(n) is the number of solutions to n = Sum_{i=1..n} c_i * prime(i) with c_i in {-1,0,1}.
%p A369560 s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
%p A369560 b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
%p A369560       b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
%p A369560     end:
%p A369560 a:= n-> b(n$2):
%p A369560 seq(a(n), n=0..32);  # _Alois P. Heinz_, Jan 25 2024
%t A369560 Table[Coefficient[Product[x^Prime[k] + 1 + 1/x^Prime[k], {k, 1, n}], x, n], {n, 0, 31}]
%Y A369560 Cf. A000040, A022894, A113040, A316706, A350404, A350695, A350880, A369390.
%K A369560 nonn
%O A369560 0,4
%A A369560 _Ilya Gutkovskiy_, Jan 25 2024