cp's OEIS Frontend

A110399 Expansion of (theta_3(q)*theta_3(q^7) - 1)/2 in powers of q.

%I A110399 #27 Nov 16 2023 07:44:09
%S A110399 1,0,0,1,0,0,1,2,1,0,2,0,0,0,0,3,0,0,0,0,0,0,2,0,1,0,0,1,2,0,0,4,0,0,
%T A110399 0,1,2,0,0,0,0,0,2,2,0,0,0,0,1,0,0,0,2,0,0,2,0,0,0,0,0,0,1,5,0,0,2,0,
%U A110399 0,0,2,2,0,0,0,0,2,0,2,0,1,0,0,0,0,0,0,4,0,0,0,2,0,0,0,0,0,0,2,1,0,0,0,0,0
%N A110399 Expansion of (theta_3(q)*theta_3(q^7) - 1)/2 in powers of q.
%C A110399 Half the number of integer solutions to x^2 + 7*y^2 = n. - _Jianing Song_, Nov 20 2019
%D A110399 Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 302, Entry 17(ii).
%F A110399 a(n) is multiplicative with a(2^e) = |e-1|, a(7^e)= 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7).
%F A110399 G.f.: Sum_{k>0} Kronecker(-7, k) x^k/(1-(-x)^k).
%F A110399 G.f.: (theta_3(q)*theta_3(q^7) - 1)/2 where theta_3(q) = 1 + 2*(q + q^4 + q^9 + ...).
%F A110399 a(2*n + 1) = A035162(2*n + 1) = A035182(2*n + 1). A033719(n) = 2*a(n) if n > 0.
%F A110399 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - _Amiram Eldar_, Nov 16 2023
%e A110399 G.f. = x + x^4 + x^7 + 2*x^8 + x^9 + 2*x^11 + 3*x^16 + 2*x^23 + ...
%t A110399 f[p_, e_] := If[MemberQ[{1, 2, 4}, Mod[p, 7]], e + 1, (1 + (-1)^e)/2]; f[2, e_] := e - 1; f[7, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 07 2023 *)
%o A110399 (PARI) {a(n) = my(x); if( n<1, 0, x = valuation(n, 2); abs(x -1) * sumdiv(n/2^x, d, kronecker(-28, d)))};
%o A110399 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, e-1,  p==7, 1, kronecker(-7, p)==-1, (1+(-1)^e)/2, e+1)))};
%o A110399 (PARI) {a(n) = my(A); if( n<1, 0, A = x *O(x^n); polcoeff( (eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-2 * eta(x^7 + A)^-2 * eta(x^14 + A)^5 * eta(x^28 + A)^-2 - 1)/2, n))};
%Y A110399 Cf. A033719 (number of integer solutions to x^2 + 7*y^2 = n).
%Y A110399 Cf. A035162, A035182.
%Y A110399 Similar sequences: A096936, A113406, A138806.
%K A110399 nonn,easy,mult
%O A110399 1,8
%A A110399 _Michael Somos_, Oct 22 2005