OFFSET
0,5
COMMENTS
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 8th equation.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(-x^5)^3 / (f(-x^10) * f(-x^2, -x^3)) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of phi(-x^5) * G(x) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 10 sequence [ 1, 0, 0, 1, -2, 1, 0, 0, 1, -1, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^(5*k^2)) / (Product_{k in Z} 1 - x^abs(5*k + 1)).
EXAMPLE
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^-1 + q^59 + q^119 + q^179 + 2*q^239 + q^359 + q^419 + 2*q^479 + q^539 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 0, -1, 2, -1, 0, 0, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^ [1, -1, 0, 0, -1, 2, -1, 0, 0, -1][k%10+1]), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 08 2015
STATUS
approved