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 A138515 Expansion of q^(-1/4) * eta(q^2)^8 / (eta(q) * eta(q^4))^2 in powers of q. 4
 1, 2, -3, -6, 2, 0, -1, 10, 0, 2, 10, -6, -7, -14, 0, 10, -12, 0, -6, 0, 9, 4, 10, 0, 18, 2, 0, -6, -14, 18, -11, -12, 0, 0, -22, 0, 20, -14, -6, -22, 0, 0, 23, 26, 0, 18, 4, 0, -14, 2, 0, 20, 0, 0, 0, -12, 3, -30, 26, 0, -30, -14, 0, 0, 2, -30, -28, 26, 0, 18, 10, 0, -13, 34, 0, 0, 20, 0, 26, -22, 0, 6, 0, -6, 18, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Number 58 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 16 2012 The weight 2 eta-quotient newform eta^8(8*z) / (eta^2(4*z)*eta^2(16*z)) appears in Theorem 2 of the Martin and Ono link in the row with conductor 64 for the strong Weil curve y^2 = x^3 - 4*x. For N(p), the number of solutions modulo primes for this elliptic curve and for y^2 = x^3 + x, see A095978. The non-vanishing p-defects p - N(p) for these two curves are given in A267859. - Wolfdieter Lang, May 26 2016 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016) LMFDB, Elliptic Curve 64.a4 (Cremona label 64a4) Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I. Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176. Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Coefficients of L-series for elliptic curve "64a4": y^2 = x^3 + x. Expansion of f(q)^2 * f(-q^2)^2 = psi(-q)^2 * phi(q)^2 = chi(q)^2 * f(-q^2)^4 = psi(q)^2 * phi(-q^2)^2 = f(q)^4 / chi(q)^2 = f(q)^6 / phi(q)^2 = f(-q^2)^6 / psi(-q)^2 = phi(q)^4 / chi(q)^6 = chi(q)^6 * psi(-q)^4 = f(q)^3 * psi(-q) = f(-q^2)^3 * phi(q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. Euler transform of period 4 sequence [2, -6, 2, -4, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i)^2 f(t) where q = exp(2 Pi i t). a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 4) with b(p) = 2 * x * (-1)^((x-1)/2) where p = x^2 + 4 * y^2. G.f.: (Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)))^2. a(n) = (-1)^n * A002171(n). a(9*n + 2) = -3 * a(n), a(9*n + 5) = a(9*n + 8) = 0. Convolution square of A138514. G.f. for{b(n)}: eta^8(8*z)/(eta^2(4*z)*eta^2(16*z)) with q = exp(2*Pi*i*z)), Im(z) > 0 (see a comment on the Martin-Ono link above). - Wolfdieter Lang, May 27 2016 EXAMPLE G.f. = 1 + 2*x - 3*x^2 - 6*x^3 + 2*x^4 - x^6 + 10*x^7 + 2*x^9 + 10*x^10 - 6*x^11 + ... G.f. for {b(n)} = q + 2*q^5 - 3*q^9 - 6*q^13 + 2*q^17 - q^25 + 10*q^29 + 2*q^37 + 10*q^41 - 6*q^45 - 7*q^49 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ -q])^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^4 / (QPochhammer[ q] QPochhammer[ q^4]))^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *) PROG (PARI) {a(n) = ellak( ellinit( [ 0, 0, 0, 1, 0], 1), 4*n + 1)}; (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)))^2, n))}; (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==1, forstep( x=1, sqrtint(p), 2, if( issquare( p - x^2), y=x; break)); y = 2 * y * (2 - (y%4)); a0 = 1; a1 = y; for(i=2, e, x = y * a1 - p * a0; a0 = a1; a1 = x); a1, if( e%2==0, (-p)^(e / 2)))))}; (Magma) A := Basis( CuspForms( Gamma0(64), 2), 342); A[1] + 2*A[3]; /* Michael Somos, May 15 2015 */ CROSSREFS Cf. A002171, A095978, A138514, A267859, A280339. Sequence in context: A260611 A263502 A002171 * A107410 A132041 A225820 Adjacent sequences: A138512 A138513 A138514 * A138516 A138517 A138518 KEYWORD sign AUTHOR Michael Somos, Mar 22 2008 STATUS approved

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Last modified October 3 16:02 EDT 2023. Contains 365868 sequences. (Running on oeis4.)