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%I A215601 #20 Feb 16 2025 08:33:18
%S A215601 1,22,-27,-18,-94,0,359,-130,0,214,-230,-594,-343,518,0,830,-396,0,
%T A215601 1098,0,729,-2068,-1670,0,594,598,0,-1746,2002,486,-1331,5148,0,0,
%U A215601 -1606,0,-2860,-3514,2538,286,0,0,-1873,-4082,0,3942,4708,0,5362,1174,0,-5060
%N A215601 Expansion of phi(-x)^2 * f(-x)^6 + 32 * x * psi(-x)^2 * f(-x^4)^6 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%C A215601 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A215601 Denoted by g_4(q) in Cynk and Hulek on page 8 as the unique weight 4 Hecke eigenform of level 32 with complex multiplication by i. - _Michael Somos_, Aug 24 2012
%C A215601 This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.
%H A215601 G. C. Greubel, <a href="/A215601/b215601.txt">Table of n, a(n) for n = 0..1000</a>
%H A215601 S. Cynk and K. Hulek, <a href="http://arXiv.org/abs/math/0509424">Construction and examples of higher-dimensional modular Calabi-Yau manifolds</a>, arXiv:math/0509424 [math.AG], 2005-2006.
%H A215601 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A215601 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A215601 Expansion of q^(-1/4) * (eta(q)^5 / eta(q^2))^2 + 32 * (eta(q) * eta(q^4)^4 / eta(q^2))^2 in powers of q.
%F A215601 a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) otherwise.
%F A215601 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^10 (t/i)^4 f(t) where q = exp(2 Pi i t).
%F A215601 a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -27 * a(n). a(n) = A215600(2*n).
%e A215601 G.f. = 1 + 22*x - 27*x^2 - 18*x^3 - 94*x^4 + 359*x^6 - 130*x^7 + 214*x^9 - 230*x^10 + ..
%e A215601 G.f. = q + 22*q^5 - 27*q^9 - 18*q^13 - 94*q^17 + 359*q^25 - 130*q^29 + 214*q^37 + ...
%t A215601 a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^5 / QPochhammer[ x^2])^2 + 32 x (QPochhammer[ x] QPochhammer[ x^4]^4 / QPochhammer[ x^2])^2, {x, 0, n}]; (* _Michael Somos_, Jan 11 2015 *)
%o A215601 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^5 / eta(x^2 + A) )^2 + 32 * x * ( eta(x + A) * eta(x^4 + A)^4 / eta(x^2 + A) )^2, n))};
%o A215601 (PARI) {a(n) = local(A, p, e, x, y, a0, a1, w=3); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p%4==3, if( e%2, 0, (-p)^(w*e/2)), y=-sum( i=0, p-1, kronecker( i^3-i, p)); a0=2; a1=y; for( i=2, w, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for( i=2, e, x=y*a1 -p^w*a0; a0=a1; a1=x); a1)))))};
%Y A215601 Cf. A000729, A002171, A008441, A215472, A215600.
%K A215601 sign
%O A215601 0,2
%A A215601 _Michael Somos_, Aug 16 2012

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