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%I #16 Mar 12 2021 22:24:48
%S 1,3,5,10,18,30,51,80,124,190,281,410,592,840,1178,1640,2253,3070,
%T 4154,5570,7422,9830,12932,16920,22028,28520,36761,47180,60280,76720,
%U 97278,122880,154693,194110,242776,302740,376424,466710,577114,711800,875707,1074790
%N Expansion of 1/q * chi(q) * chi(q^5) * chi(-q^20)^2 / chi(-q)^2 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 G. C. Greubel, <a href="/A298116/b298116.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 * f(q) * f(q^5) / (phi(-q) * psi(q^10)) in powers of q where f(), phi(), psi() are Ramanujan theta functions.
%F Euler transform of period 20 sequence [3, -1, 3, 0, 4, -1, 3, 0, 3, -4, 3, 0, 3, -1, 4, 0, 3, -1, 3, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A058555(n) = A298107(n) unless n=0.
%F Expansion of (eta(q^2) * eta(q^10))^4/(eta(q^4)*eta(q^5)*(eta(q)* eta(q^20))^3) in powers of q. - _G. C. Greubel_, Mar 20 2018
%F a(n) ~ exp(2*Pi*sqrt(n/5)) / (2*5^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Mar 21 2018
%e G.f. = q^-1 + 3 + 5*q + 10*q^2 + 18*q^3 + 30*q^4 + 51*q^5 + 80*q^6 + 124*q^7 + ...
%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^10, q^20]^2 QPochhammer[-q, q]^2 QPochhammer[-q, q^2] QPochhammer[-q^5, q^10], {q, 0, n}];
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[(eta[q^2]* eta[q^10])^4/(eta[q^4]*eta[q^5]*(eta[q]*eta[q^20])^3), {q,0,n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 20 2018 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^10 + A)^4 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)^3), n))};
%Y Essentially the same as A058555 and A298107.
%K nonn
%O -1,2
%A _Michael Somos_, Jan 12 2018
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