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A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

%I A132969 #23 Feb 16 2025 08:33:06
%S A132969 1,3,2,1,5,5,3,5,6,10,10,8,13,15,15,16,23,27,25,30,35,40,42,45,55,66,
%T A132969 68,70,86,95,100,110,125,141,150,161,185,207,215,235,266,293,310,335,
%U A132969 375,410,438,470,521,575,610,653,725,785,835,900,983,1070,1140
%N A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.
%C A132969 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A132969 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; top of p. 60.
%H A132969 G. C. Greubel, <a href="/A132969/b132969.txt">Table of n, a(n) for n = 0..1000</a>
%H A132969 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/323.pdf">Integer partitions with even parts below odd parts and the mock theta functions</a>, to appear in Annals of Combinatorics, 2017.
%H A132969 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132969 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132969 Expansion of phi(q) + 2 * psi(q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
%F A132969 Expansion of q^(1/24) * eta(q^2)^7 / (eta(q) * eta(q^4))^3 in powers of q.
%F A132969 Euler transform of period 4 sequence [ 3, -4, 3, -1, ...].
%F A132969 G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t).
%F A132969 G.f.: ( Sum_{k in Z} x^k^2 ) * ( Product_{k>0} (1 + x^(2*k-1)) ).
%F A132969 G.f.: Product_{k>0} (1 - x^(2*k)) * ((1 + x^k) / (1 + x^(2*k)))^3.
%F A132969 a(n) = (-1)^n * A132970(n). a(n) = (-1)^n * A124226(n) unless n=1.
%F A132969 a(n) ~ exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - _Vaclav Kotesovec_, Sep 08 2015
%e A132969 G.f. = 1 + 3*x + 2*x^2 + x^3 + 5*x^4 + 5*x^5 + 3*x^6 + 5*x^7 + 6*x^8 + 10*x^9 + ...
%e A132969 G.f. = 1/q + 3*q^23 + 2*q^47 + q^71 + 5*q^95 + 5*q^119 + 3*q^143 + 5*q^167 +...
%t A132969 nmax = 60; CoefficientList[Series[Product[(1 - x^(2*k)) * ( (1 + x^k) / (1 + x^(2*k)) )^3, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%t A132969 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2], {x, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)
%o A132969 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + x*O(x^n)) * sum(k=1, sqrtint(n), 2 * x^k^2, 1), n))};
%o A132969 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A) * eta(x^4 + A))^3, n))};
%Y A132969 Cf. A053250, A053251, A124226, A132970.
%K A132969 nonn
%O A132969 0,2
%A A132969 _Michael Somos_, Sep 04 2007