A259357 Expansion of f(-x^5)^2 / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.

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

Author

Michael Somos, Jun 24 2015

Keywords

• nonn

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).

Examples

			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 + ...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 3.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

Crossrefs

Cf. A035959, A113428, A259358.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];

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))};

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.