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Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.
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%I #55 Mar 12 2021 22:24:42

%S 1,-1,1,-2,2,-3,4,-5,6,-8,10,-12,15,-18,22,-27,32,-38,46,-54,64,-76,

%T 89,-104,122,-142,165,-192,222,-256,296,-340,390,-448,512,-585,668,

%U -760,864,-982,1113,-1260,1426,-1610,1816,-2048,2304,-2590,2910,-3264,3658,-4097,4582,-5120,5718,-6378

%N Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.

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

%C Number of partitions of n into distinct parts with an even number of odd parts minus partitions of n into distinct parts with an odd number of odd parts. G.f.: Product_{i=1..oo} (1+(-1)^i*x^i). - _Jon Perry_, Jun 04 2004

%H G. C. Greubel, <a href="/A081360/b081360.txt">Table of n, a(n) for n = 0..1000</a>

%H Jason Fulman, <a href="http://dx.doi.org/10.1090/S0273-0979-01-00920-X">Random matrix theory over finite fields</a>, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=0. - _N. J. A. Sloane_, Aug 31 2014

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.

%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>

%H E. W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EllipticLambdaFunction.html">Elliptic Lambda Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>

%F Expansion of 1 / chi(x) = chi(-x) / chi(-x^2) = f(x) / phi(x) = f(-x) / phi(-x^2) = psi(-x) / f(-x^2) = f(-x^2) / f(x) = f(-x^4) / psi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

%F Expansion of (lambda * (1 - lambda) / (16 * q))^(1/24) in powers of q where lambda is a modular elliptic function and q = exp(Pi i z) is the nome. - _Michael Somos_, Jul 19 2012

%F Expansion of q^(-1/24) * eta(q) * eta(q^4) / eta(q^2)^2 in powers of q.

%F Expansion of q^(-1/24) / f(t) in powers of q = exp(Pi i t) where f() is Weber's function.

%F Euler transform of period 4 sequence [-1, 1, -1, 0, ...].

%F Given g.f. A(x), B(x) = x * A(x^3)^8 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (u - v^2) * (v - u^2) - (4 * u * v * (1 - u*v))^2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jul 16 2007

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

%F a(n) = (-1)^n * A000009(n). Convolution inverse of A000700.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015

%F G.f.: (1/2)*(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Nov 21 2016

%F G.f.: exp(-Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 08 2018

%F Given g.f. A(x), B(x) = 2^(1/4) * x * A(x^24) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^6 + v^6 + 2*u*v * ((u*v)^4 - 1). - _Michael Somos_, Mar 14 2019

%e G.f. = 1 - x + x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 5*x^7 + 6*x^8 - 8*x^9 + ...

%e G.f. = q - q^25 + q^49 - 2*q^73 + 2*q^97 - 3*q^121 + 4*q^145 - 5*q^169 + ...

%p read theta; t1:=series(eta,q,48); t2:= q^(-1/24)*t1*subs(q=q^4,t1)/subs(q=q^2,t1)^2; series(t2,q,48); seriestolist(%); # _N. J. A. Sloane_, Aug 24 2007

%t a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x^2], {x, 0, n}]; (* _Michael Somos_, Jul 19 2012 *)

%t a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}]; (* _Michael Somos_, Jul 19 2012 *)

%t a[ n_] := SeriesCoefficient[ With[ {m = ModularLambda[ Log[ q] / (Pi I)]}, ( m (1 - m) / (16 q))^(1/24)], {q, 0, n}]; (* _Michael Somos_, Jul 19 2012 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x], {x, 0, n}]; (* _Michael Somos_, Nov 22 2016 *)

%t nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 30 2015 *)

%t (QPochhammer[-1, -x]/2 + O[x]^60)[[3]] (* _Vladimir Reshetnikov_, Nov 21 2016 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2, n))};

%Y Cf. A000009, A000700, A278428.

%K sign

%O 0,4

%A _Michael Somos_, Mar 18 2003