OFFSET
0,4
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Fulman, Random matrix theory over finite fields, 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
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
E. W. Weisstein's World of Mathematics, Elliptic Lambda Function
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
FORMULA
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.
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
Expansion of q^(-1/24) * eta(q) * eta(q^4) / eta(q^2)^2 in powers of q.
Expansion of q^(-1/24) / f(t) in powers of q = exp(Pi i t) where f() is Weber's function.
Euler transform of period 4 sequence [-1, 1, -1, 0, ...].
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.
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
G.f.: Product_{k>0} 1 / ( 1 + x^(2k - 1)) = Product_{k>0} (1 + (-x)^k).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
G.f.: (1/2)*(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 21 2016
G.f.: exp(-Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018
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
EXAMPLE
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 + ...
G.f. = q - q^25 + q^49 - 2*q^73 + 2*q^97 - 3*q^121 + 4*q^145 - 5*q^169 + ...
MAPLE
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
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
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 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
(QPochhammer[-1, -x]/2 + O[x]^60)[[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
PROG
(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))};
KEYWORD
sign
AUTHOR
Michael Somos, Mar 18 2003
STATUS
approved