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Expansion of chi(x) * psi(x^6) * phi(-x^30) / (f(-x^4) * psi(x^5)) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
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%I #9 Mar 12 2021 22:24:48

%S 1,1,0,1,2,1,1,3,4,4,4,6,8,8,8,11,16,17,17,23,31,32,32,42,54,56,59,77,

%T 94,99,106,129,156,167,178,214,257,276,295,350,416,445,474,559,652,

%U 698,752,877,1012,1089,1174,1349,1542,1662,1792,2042,2327,2512,2706

%N Expansion of chi(x) * psi(x^6) * phi(-x^30) / (f(-x^4) * psi(x^5)) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

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

%H Vaclav Kotesovec, <a href="/A262175/b262175.txt">Table of n, a(n) for n = 0..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 q^(1/12) * eta(q^2)^2 * eta(q^5) * eta(q^12)^2 * eta(q^30)^2 / (eta(q) * eta(q^4)^2 * eta(q^6) * eta(q^10)^2 * eta(q^60)) in powers of q.

%F Euler transform of a period 60 sequence.

%F a(n) = A139632(3*n).

%F a(n) ~ exp(Pi*sqrt(3*n/10)) / (2^(5/4) * 3^(3/4) * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017

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

%e G.f. = q^-1 + q^11 + q^35 + 2*q^47 + q^59 + q^71 + 3*q^83 + 4*q^95 + ...

%t a[ n_] := SeriesCoefficient[ x^(-1/8) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^3] EllipticTheta[ 4, 0, x^30] / (QPochhammer[ x^4] EllipticTheta[ 2, 0, x^(5/2)]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^5 + A) * eta(x^12 + A)^2 * eta(x^30 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^10 + A)^2 * eta(x^60 + A)), n))};

%Y Cf. A139632.

%K nonn

%O 0,5

%A _Michael Somos_, Sep 13 2015