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Expansion of chi(q) * chi(q^2) / (chi(q^5) * chi(q^10)) in powers of q where chi() is a Ramanujan theta function.
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%I #11 Mar 12 2021 22:24:44

%S 1,1,1,2,1,1,2,2,2,4,4,4,5,5,6,6,8,9,10,12,14,15,17,20,21,23,26,30,32,

%T 37,42,44,50,56,60,66,74,80,88,98,109,119,130,144,154,167,184,200,218,

%U 241,262,284,308,334,362,390,426,462,498,542,589,633,685,742,796,858

%N Expansion of chi(q) * chi(q^2) / (chi(q^5) * chi(q^10)) 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 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 Euler transform of period 40 sequence [ 1, 0, 1, -1, 0, 0, 1, 0, 1, 0, 1, -1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1, 0, 1, 0, ...].

%F Given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (u^2 + v^2).

%F G.f.: Product_{k>0} (1 + x^k) * (1 + x^(20*k)) / ( (1 + x^(4*k)) * (1+x^(5*k))).

%F Convolution inverse of A128763.

%F a(n) ~ exp(Pi*sqrt(n/5)) / (2^(3/2) * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015

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

%e G.f. = q + q^3 + q^5 + 2*q^7 + q^9 + q^11 + 2*q^13 + 2*q^15 + 2*q^17 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5, - x^5] QPochhammer[ x^10, -x^10]) / (QPochhammer[ x, -x] QPochhammer[ x^2, -x^2]), {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t nmax = 40; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(20*k)) / ( (1 + x^(4*k)) * (1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^8 + A) * eta(x^10 + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^40 + A)), n))};

%Y Cf. A128763.

%K nonn

%O 0,4

%A _Michael Somos_, Mar 25 2007