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Expansion of (1/q) * phi(-q) * phi(q^5) / (f(-q^4) * f(-q^20)) in powers of q where phi(), f() are Ramanujan theta functions.
2

%I #12 Mar 12 2021 22:24:48

%S 1,-2,0,0,3,0,-4,0,4,0,-4,0,7,0,-12,0,13,0,-16,0,22,0,-28,0,38,0,-44,

%T 0,55,0,-72,0,83,0,-104,0,129,0,-156,0,187,0,-220,0,273,0,-328,0,384,

%U 0,-452,0,539,0,-652,0,757,0,-880,0,1041,0,-1220,0,1428,0

%N Expansion of (1/q) * phi(-q) * phi(q^5) / (f(-q^4) * f(-q^20)) in powers of q where phi(), f() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A298203/b298203.txt">Table of n, a(n) for n = -1..1000</a>

%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 (1/q) * chi(q) * chi(-q)^3 * chi(q^5)^3 * chi(-q^5) in powers of q where chi() is a Ramanujan theta function.

%F Expansion of eta(q)^2 * eta(q^10)^5 / (eta(q^2) * eta(q^4)* eta(q^5)^2 * eta(q^20)^3) in powers of q.

%F Euler transform of period 20 sequence [-2, -1, -2, 0, 0, -1, -2, 0, -2, -4, -2, 0, -2, -1, 0, 0, -2, -1, -2, 0, ...].

%F a(2*n) = 0 except n=0. a(2*n + 1) = A058559(n) for all n in Z.

%e G.f. = q^-1 - 2 + 3*q^3 - 4*q^5 + 4*q^7 - 4*q^9 + 7*q^11 - 12*q^13 + 13*q^15 + ...

%t a[ n_] := SeriesCoefficient[ 1/q EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^5] / (QPochhammer[ q^4] QPochhammer[ q^20]), {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q, q^2] QPochhammer[ q, q^2]^3 QPochhammer[ -q^5, q^10]^3 QPochhammer[ q^5, q^10], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^10 + A)^5 / (eta(x^2 + A) * eta(x^4 + A)* eta(x^5 + A)^2 * eta(x^20 + A)^3), n))};

%Y Cf. A058559.

%K sign

%O -1,2

%A _Michael Somos_, Jan 14 2018