A116418 - OEIS

%I #32 Sep 08 2022 08:45:24

%S 1,-2,-4,6,8,4,-16,-24,7,8,44,18,-34,-12,-40,24,-33,-16,72,-6,50,-8,8,

%T -24,-16,32,-76,-66,-54,48,-32,120,176,-14,-28,-54,56,-16,-72,-48,

%U -167,-88,92,48,64,-36,152,72,18,68,-148,96,-82,24,56,-168,-105,80,-28,-18,-232,-48,216,-96,206,66,20,-42,198,32

%N Expansion of q^(-1/3) * b(q) * c(q) * b(q^2) / 3 in powers of q where b(), c() are cubic AGM theta functions.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H Seiichi Manyama, <a href="/A116418/b116418.txt">Table of n, a(n) for n = 0..1000</a>

%H LMFDB, <a href="http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/18/3/17/a/">Newform 18.3.17.a</a>.

%F Expansion of q^(-1/3) * b(q) * c(q) * b(q^2) / 3 in powers of q where b(), c() are cubic AGM theta functions.

%F Expansion of q^(-1/3) * eta(q)^2 * eta(q^2)^3 * eta(q^3)^2 / eta(q^6) in powers of q.

%F Euler transform of period 6 sequence [ -2, -5, -4, -5, -2, -6, ...].

%F Expansion of a newform level 18 weight 3 and character chi_18(17,.).

%F a(n) = b(3*n + 1) 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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 3888^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208384.

%F G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^3 * (1 - x^(3*k))^2 / (1 - x^(6*k)).

%F a(n) = A208384(3*n + 1) = A208385(3*n + 1).

%F a(2*n + 1) = -2 * A122407(n). - _Michael Somos_, Mar 30 2015

%F a(4*n + 1) = -2 * a(n). a(4*n + 3) = 6 * A280822(n). - _Michael Somos_, Jan 08 2017

%e G.f. = 1 - 2*x - 4*x^2 + 6*x^3 + 8*x^4 + 4*x^5 - 16*x^6 - 24*x^7 + 7*x^8 + ...

%e G.f. = q - 2*q^4 - 4*q^7 + 6*q^10 + 8*q^13 + 4*q^16 - 16*q^19 - 24*q^22 +...

%t s[n_] := Series[Product[(1-x^k)^2*(1-x^(2*k))^3*(1-x^(3*k))^2/(1-x^(6*k)), {k, 1, n}], {x, 0, n}] // Normal; a[k_] := SeriesCoefficient[s[n], {x, 0, k}]; a[0]=1; Table[a[n], {n, 0, 69}] (* _Jean-François Alcover_, Feb 04 2014 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^2]^3 QPochhammer[ x^3]^2 / QPochhammer[ x^6], {x, 0, n}]; (* _Michael Somos_, Mar 30 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A)^3 * eta(x^3 + A)^2 / eta(x^6 + A), n))};

%o (Magma) A := Basis( ModularForms( Gamma1(18), 3), 210); A[2] - 2*A[5] - 4*A[8] + 6*A[11] + 8*A[14] + 4*A[17] - 16*A[20] - 24*A[23]; /* _Michael Somos_, Mar 30 2015 */

%o (Magma) A := Basis( CuspForms( Gamma1(18), 3), 210); A[1] -2*A[4] - 4*A[7] + 6*A[10]; /* _Michael Somos_, Jan 08 2017 */

%Y Cf. A122407, A208384, A208385, A280822.

%K sign

%O 0,2

%A _Michael Somos_, Feb 13 2006