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A259357
Expansion of f(-x^5)^2 / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.
2
1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 5, 5, 5, 6, 7, 7, 9, 9, 11, 11, 13, 13, 16, 17, 19, 20, 23, 24, 27, 29, 32, 34, 38, 40, 46, 48, 52, 56, 62, 65, 72, 76, 84, 89, 97, 102, 113, 119, 129, 137, 149, 157, 171, 181, 196, 208, 224, 236, 256, 270, 290, 308, 331
OFFSET
0,5
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 3.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
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))).
Convolution of A035959 and A113428.
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