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

0,5

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 3.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

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))).

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

Cf. A035959, A113428, A259358.

Sequence in context: A158925 A262868 A342097 * A031265 A029202 A173894

Adjacent sequences: A259354 A259355 A259356 * A259358 A259359 A259360

Keyword

nonn

Author

Michael Somos, Jun 24 2015

Status

approved