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A246832 Expansion of psi(x) * psi(x^2) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A246832 #10 Feb 16 2025 08:33:23
%S A246832 1,1,3,4,2,5,2,3,7,5,5,6,5,3,10,6,3,7,7,4,11,9,3,14,8,8,5,4,7,10,13,7,
%T A246832 8,10,7,15,8,4,17,10,8,11,10,5,16,11,4,11,15,8,14,10,7,22,8,9,14,8,13,
%U A246832 21,12,5,13,15,6,21,15,9,13,8,11,9,12,15,20,21
%N A246832 Expansion of psi(x) * psi(x^2) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A246832 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246832 G. C. Greubel, <a href="/A246832/b246832.txt">Table of n, a(n) for n = 0..2500</a>
%H A246832 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246832 Expansion of psi(x) * psi(x^2)^3 / psi(x^4) in powers of x where psi() is a Ramanujan theta function.
%F A246832 Expansion of q^(-3/8) * eta(q^4)^7 / (eta(q) * eta(q^2) * eta(q^8)^2) in powers of q.
%F A246832 Euler transform of period 8 sequence [1, 2, 1, -5, 1, 2, 1, -3, ...].
%F A246832 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 2 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246816.
%e A246832 G.f. = 1 + x + 3*x^2 + 4*x^3 + 2*x^4 + 5*x^5 + 2*x^6 + 3*x^7 + 7*x^8 + ...
%e A246832 G.f. = q^3 + q^11 + 3*q^19 + 4*q^27 + 2*q^35 + 5*q^43 + 2*q^51 + 3*q^59 + ...
%t A246832 eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-3/8)* eta[q^4]^7/(eta[q]*eta[q^2]*eta[q^8]^2), {q, 0, 60}], q]]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Aug 05 2018 *)
%o A246832 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^7 / (eta(x + A) * eta(x^2 + A) * eta(x^8 + A)^2), n))};
%Y A246832 Cf. A246816.
%K A246832 nonn
%O A246832 0,3
%A A246832 _Michael Somos_, Sep 04 2014