A182004 - OEIS

%I #23 Sep 07 2018 03:08:48

%S 1,0,1,2,-2,0,0,-2,0,2,2,0,1,2,0,-2,0,0,-2,0,1,0,2,0,-2,-2,0,-2,-2,2,

%T 1,0,0,0,-2,0,0,-2,-2,2,0,0,3,2,0,-2,0,0,-2,2,0,0,0,0,0,-4,0,-2,-2,0,

%U 2,0,0,0,-2,-2,0,-2,0,2,2,0,3,2,0,0,0,0,-2,2,0

%N Expansion of q^(-1/4) * (eta(q^4) * eta(q^25) + eta(q) * eta(q^100))^2 / (eta(q^2) * eta(q^50)) in powers of q.

%C In Koehler, page 212 is an example 1 defining f = f_1 + 2f_{13} + f_{25} whose q-expansion is this sequence.

%H G. C. Greubel, <a href="/A182004/b182004.txt">Table of n, a(n) for n = 0..2500</a>

%H G. Köhler, <a href="https://doi.org/10.1007/s000130050561">Eta products of weight 1 and level 36</a>, Arch. Math. 76 (2001) 202-214.

%F a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = b(5^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4), b(p^e) = (e + 1) * s^e where s = Kronecker(10, p) for other primes p.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (1600 t)) = 40 (t/i) f(t) where q = exp(2 Pi i t).

%F G.f.: (1/4) * Sum_{i, j in Z} Kronecker(10, i^2 + j^2) * x^(i^2 + j^2).

%e G.f. = 1 + x^2 + 2*x^3 - 2*x^4 - 2*x^7 + 2*x^9 + 2*x^10 + x^12 + 2*x^13 + ...

%e G.f. = q + q^9 + 2*q^13 - 2*q^17 - 2*q^29 + 2*q^37 + 2*q^41 + q^49 + 2*q^53 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/4)* (eta[q^4]*eta[q^25] + eta[q]*eta[q^100])^2/(eta[q^2]*eta[q^50]), {q, 0, 50}], q] (* _G. C. Greubel_, Aug 11 2018 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^25 + A) + x^3 * eta(x + A) * eta(x^100 + A))^2 / (eta(x^2 + A) * eta(x^50 + A)), n))};

%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor( 4*n + 1 ); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2 || p==5, 0, p%4==3, (1 + (-1)^e) / 2, (e+1) * kronecker( 10, p) ^ e )))};

%K sign

%O 0,4

%A _Michael Somos_, Apr 06 2012