A259358 Expansion of f(-x^5)^2 / f(-x^2, -x^3) in powers of x where f(,) is the Ramanujan general theta function.

1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 3, 2, 4, 4, 5, 4, 6, 5, 7, 7, 8, 8, 11, 10, 12, 13, 15, 14, 18, 17, 21, 21, 24, 25, 29, 29, 34, 35, 39, 40, 47, 47, 53, 55, 61, 63, 72, 73, 82, 86, 94, 97, 109, 112, 124, 129, 141, 147, 162, 167, 183, 192, 208, 217, 237

Offset

0,7

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 (2.1.22).

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 4.

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

Expansion of f(-x^5) * H(x) in powers of x where f() is a Ramanujan theta funcation and H() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015

Euler transform of period 5 sequence [ 0, 1, 1, 0, -1, ...].

G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-3)) * (1 - x^(5*k-2))).

Convolution of A035959 and A113429.

Example

G.f. = 1 + x^2 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q^47 + q^287 + q^407 + q^527 + 2*q^767 + q^887 + 2*q^1007 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, 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, 0, -1, -1, 0][k%5+1]), n))};

Crossrefs

Cf. A035959, A113429, A259357.

Sequence in context: A241065 A086376 A160089 * A290086 A129363 A308342

Adjacent sequences: A259355 A259356 A259357 * A259359 A259360 A259361

Keyword

nonn

Author

Michael Somos, Jun 24 2015

Status

approved