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Expansion of (chi(-x) * chi(-x^19))^(-2) in powers of x where chi() is a Ramanujan theta function.
1

%I #12 Mar 12 2021 22:24:45

%S 1,2,3,6,9,14,22,32,46,66,93,128,176,238,319,426,562,736,960,1244,

%T 1602,2054,2620,3324,4203,5292,6634,8290,10322,12808,15845,19542,

%U 24028,29468,36042,43966,53506,64960,78685,95106,114709,138066,165855,198856,237979

%N Expansion of (chi(-x) * chi(-x^19))^(-2) in powers of x 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 Michael 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 q^(-5/3) * (eta(q^2) * eta(q^38) / (eta(q) * eta(q^19)))^2 in powers of q.

%F Euler transform of period 38 sequence [ 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (342 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134005.

%F G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(19*k)))^2.

%F Convolution inverse of A134005.

%F a(n) ~ exp(2*Pi*sqrt(10*n/57)) * 5^(1/4) / (2^(11/4) * 57^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015

%e G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...

%e G.f. = q^5 + 2*q^8 + 3*q^11 + 6*q^14 + 9*q^17 + 14*q^20 + 22*q^23 + 32*q^26 + ...

%t nmax = 40; CoefficientList[Series[Product[((1 + x^k) * (1 + x^(19*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^19, x^19])^2, {x, 0, n}]; (* _Michael Somos_, Oct 30 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^38 + A) / (eta(x + A) * eta(x^19 + A)))^2, n))};

%Y Cf. A134005.

%K nonn

%O 0,2

%A _Michael Somos_, Oct 01 2007