A292698 - OEIS

%I #24 Apr 30 2022 07:58:17

%S 0,0,0,1,2,8,27,83,292,944,3279,11291,38992,138066,490248,1757360,

%T 6347321,22998089,83780199,306819363,1128999790,4174251748,

%U 15507225620,57767819903,215327188611,803901214851,3013081897103,11331883386143,42737620941612

%N Number of solutions to 3 +- 7 +- 13 +- 19 +- ... +- prime(4*n) = 0.

%H Seiichi Manyama, <a href="/A292698/b292698.txt">Table of n, a(n) for n = 1..200</a>

%F Constant term in the expansion of 1/2 * Product_{k=1..2*n} (x^prime(2*k) + 1/x^prime(2*k)).

%e For n = 4 the solution is 3 - 7 - 13 + 19 - 29 + 37 + 43 - 53 = 0.

%p s:= proc(n) s(n):= `if`(n=0, 0, ithprime(2*n)+s(n-1)) end:

%p b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,

%p       (p-> b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(2*i))))

%p     end:

%p a:= n-> b(0, 2*n)/2:

%p seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 21 2017

%t s[n_] := s[n] = If[n == 0, 0, Prime[2n] + s[n-1]];

%t b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,

%t      With[{p = Prime[2i]}, b[n+p, i-1] + b[Abs[n-p], i-1]]]];

%t a[n_] := b[0, 2n]/2;

%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Apr 30 2022, after _Alois P. Heinz_ *)

%o (PARI) {a(n) = 1/2*polcoeff(prod(k=1, 2*n, x^prime(2*k)+1/x^prime(2*k)), 0)}

%Y Cf. A000040, A022894, A083309, A292699, A292700.

%K nonn

%O 1,5

%A _Seiichi Manyama_, Sep 21 2017