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A132970 Expansion of phi(-x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A132970 #15 Feb 16 2025 08:33:06
%S A132970 1,-3,2,-1,5,-5,3,-5,6,-10,10,-8,13,-15,15,-16,23,-27,25,-30,35,-40,
%T A132970 42,-45,55,-66,68,-70,86,-95,100,-110,125,-141,150,-161,185,-207,215,
%U A132970 -235,266,-293,310,-335,375,-410,438,-470,521,-575,610,-653,725,-785,835,-900,983,-1070,1140,-1220,1331
%N A132970 Expansion of phi(-x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A132970 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A132970 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 60, Eqs. (26.64),(26.65),(26.66)
%H A132970 G. C. Greubel, <a href="/A132970/b132970.txt">Table of n, a(n) for n = 0..1000</a>
%H A132970 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132970 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132970 Expansion of phi(-q) + 2 * psi(-q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
%F A132970 Expansion of q^(1/24) * eta(q)^3 / eta(q^2)^2 in powers of q.
%F A132970 Euler transform of period 2 sequence [ -3, -1, ...].
%F A132970 G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 48^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A085140.
%F A132970 G.f.: ( Sum_{k in Z} (-1)^k * x^k^2 ) / ( Product_{k>0} (1 + x^k) ).
%F A132970 G.f.: Product_{k>0} (1 - x^k) / (1 + x^k)^2.
%F A132970 a(n) = (-1)^n * A132969(n). a(n) = A124226(n) unless n=1.
%F A132970 a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - _Vaclav Kotesovec_, Oct 14 2017
%e A132970 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 + ...
%e A132970 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 A132970 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* _Michael Somos_, Jul 20 2015 *)
%o A132970 (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 * (-1)^k * x^k^2, 1), n))};
%o A132970 (PARI) {a(n) = my(A) ; if( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^3 / eta(x^2 + A)^2, n))};
%Y A132970 CF. A124226, A132969.
%K A132970 sign
%O A132970 0,2
%A A132970 _Michael Somos_, Sep 04 2007