A132970 - OEIS

%I #14 Mar 12 2021 22:24:44

%S 1,-3,2,-1,5,-5,3,-5,6,-10,10,-8,13,-15,15,-16,23,-27,25,-30,35,-40,

%T 42,-45,55,-66,68,-70,86,-95,100,-110,125,-141,150,-161,185,-207,215,

%U -235,266,-293,310,-335,375,-410,438,-470,521,-575,610,-653,725,-785,835,-900,983,-1070,1140,-1220,1331

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

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

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 60, Eqs. (26.64),(26.65),(26.66)

%H G. C. Greubel, <a href="/A132970/b132970.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 phi(-q) + 2 * psi(-q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.

%F Expansion of q^(1/24) * eta(q)^3 / eta(q^2)^2 in powers of q.

%F Euler transform of period 2 sequence [ -3, -1, ...].

%F 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 G.f.: ( Sum_{k in Z} (-1)^k * x^k^2 ) / ( Product_{k>0} (1 + x^k) ).

%F G.f.: Product_{k>0} (1 - x^k) / (1 + x^k)^2.

%F a(n) = (-1)^n * A132969(n). a(n) = A124226(n) unless n=1.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - _Vaclav Kotesovec_, Oct 14 2017

%e 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 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 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* _Michael Somos_, Jul 20 2015 *)

%o (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 (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 CF. A124226, A132969.

%K sign

%O 0,2

%A _Michael Somos_, Sep 04 2007