OFFSET
0,5
COMMENTS
REFERENCES
G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, 2012, see p. 12, Entry 2.1.3, Equation (2.1.21).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 3.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(-x^5) * f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(-x^5) * G(x) in powers of x where f() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 5 sequence [ 1, 0, 0, 1, -1, ...].
G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-4)) * (1 - x^(5*k-1))).
EXAMPLE
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + ...
G.f. = q^23 + q^143 + q^263 + q^383 + 2*q^503 + q^623 + 2*q^743 + 2*q^863 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {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][k%5+1]), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 24 2015
STATUS
approved