OFFSET
0,10
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 36, Eq. (4.11). MR0858826 (88b:11063).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(-x^2) * f(-x^5) / (f(-x^4) * f(-x, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of (f(-x^13, -x^17) + x * f(-x^7, -x^23)) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [ 1, -1, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, ...].
G.f.: Sum_{k>=0} x^k^2 / (Product_{j=1..k} 1 - x^(4*j)).
G.f.: (Sum_{k in Z} (-1)^k^2 * x^(k * (5*k + 1) / 2)) / (Sum_{k in Z} (-1)^k^2 * x^(k * (2*k + 1))). - Michael Somos, Oct 08 2015
EXAMPLE
G.f. = 1 + x + x^4 + x^5 + x^8 + 2*x^9 + 2*x^12 + 2*x^13 + 3*x^16 + 3*x^17 + ...
G.f. = 1/q + q^9 + q^39 + q^49 + q^79 + 2*q^89 + 2*q^119 + 2*q^129 + 3*q^159 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / (QPochhammer[ x^4] QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2) / QPochhammer[ x^4, x^4, k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := SeriesCoefficient[ Sqrt[2] x^(1/8) QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5] / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, k, 1 - x^(4*i), 1 + x * O(x^(n - k^2)))), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Feb 04 2008
STATUS
approved