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A132319 Expansion of q^-1 * (chi(-q) * chi(-q^7))^3 in powers of q where chi() is a Ramanujan theta function. 3

%I

%S 1,-3,3,-4,9,-12,15,-24,39,-52,66,-96,130,-168,219,-292,390,-492,625,

%T -804,1023,-1280,1599,-2016,2508,-3096,3807,-4688,5760,-7020,8532,

%U -10368,12585,-15156,18213,-21912,26287,-31404,37410,-44584,53004,-62784,74245,-87768

%N Expansion of q^-1 * (chi(-q) * chi(-q^7))^3 in powers of q where chi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H Seiichi Manyama, <a href="/A132319/b132319.txt">Table of n, a(n) for n = -1..10000</a>

%H M. Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (eta(q) * eta(q^7) / (eta(q^2) * eta(q^14)))^3 in powers of q.

%F Euler transform of period 14 sequence [ -3, 0, -3, 0, -3, 0, -6, 0, -3, 0, -3, 0, -3, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 8 * u + 6 * u * v.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A120006.

%F G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(7*k)))^-3.

%F a(n) = A058503(n) unless n = 0. Convolution inverse is A120006.

%F a(n) = -(-1)^n * exp(2*Pi*sqrt(n/7)) / (2*7^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%e G.f. = 1/q - 3 + 3*q - 4*q^2 + 9*q^3 - 12*q^4 + 15*q^5 - 24*q^6 + 39*q^7 - ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q, q^2] QPochhammer[ q^7, q^14])^3 / q, {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)))^3, n))};

%Y Cf. A058503, A120006.

%K sign

%O -1,2

%A _Michael Somos_, Aug 18 2007

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Last modified August 10 08:22 EDT 2020. Contains 336368 sequences. (Running on oeis4.)