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 A122407 Expansion of q^(-2/3) * b(q) * b(q^2) * c(q^2) / 3 in powers of q where b(), c() are cubic AGM theta functions. 4
 1, -3, -2, 12, -4, -9, 6, -12, 8, 3, 4, 12, -16, 33, -24, -60, 7, 27, 8, 24, 44, -24, 18, -36, -34, -48, -12, 84, -40, 9, 24, 48, -33, 21, -16, 0, 72, -111, -6, 36, 50, -96, -8, -120, 8, 111, -24, 96, -16, 195, 32, -132, -76, 48, -66, 24, -54, -123, 48, -144, -32, 102, 120, -24, 176, 117, -14, -12, -28, -192, -54 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 N. Elkies, Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418, MathOverflow. LMFDB, Newform 18.3.17.a. FORMULA Expansion of q^(-2/3) * eta(q)^3 * eta(q^2)^2 * eta(q^6)^2 / eta(q^3) in powers of q. Euler transform of period 6 sequence [-3, -5, -2, -5, -3, -6, ...]. Expansion of a newform level 18 weight 3 and character chi_18(17,.). - Michael Somos, Jan 08 2017 a(n) = sqrt(-2) * b(3*n + 2) where b() is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - Kronecker(-12, p) * p^2 * b(p^(e-2)). - Michael Somos, Jan 08 2017 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 972^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208385. G.f.: Product_{k>0} (1 - x^k)^6 * (1 + x^k)^4 * (1 - x^k + x^(2*k))^2  *(1 + x^k + x^(2*k)). a(n) = A208384(3*n + 2). -2 * a(n) = A208385(3*n + 2). -2 * a(n) = A116418(2*n + 1) = a(4*n + 2). a(2*n) = A116418(n). - Michael Somos, Jan 08 2017 EXAMPLE G.f. = 1 - 3*x - 2*x^2 + 12*x^3 - 4*x^4 - 9*x^5 + 6*x^6 - 12*x^7 + 8*x^8 + ... G.f. = q^2 - 3*q^5 - 2*q^8 + 12*q^11 - 4*q^14 - 9*q^17 + 6*q^20 - 12*q^23 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^2]^2 QPochhammer[ x^6]^2 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jan 08 2017 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^2 + A)^2 * eta(x^6 + A)^2 / eta(x^3 + A), n))}; (MAGMA) A := Basis( CuspForms( Gamma1(18), 3), 211); A[2] - 3*A[5] - 2*A[8]; /* Michael Somos, Jan 08 2017 */ CROSSREFS Cf. A116418, A208384, A208385. Sequence in context: A178384 A078563 A016560 * A232752 A195200 A098646 Adjacent sequences:  A122404 A122405 A122406 * A122408 A122409 A122410 KEYWORD sign AUTHOR Michael Somos, Sep 02 2006 STATUS approved

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Last modified June 5 23:10 EDT 2020. Contains 334858 sequences. (Running on oeis4.)