A259910 Expansion of f(-x^2, -x^3)^3 / f(-x)^2 in powers of x where f(,) is the Ramanujan general theta function.

1, 2, 2, 1, 2, 3, 4, 3, 3, 4, 6, 6, 7, 8, 8, 9, 11, 12, 13, 14, 17, 19, 21, 21, 25, 27, 30, 31, 35, 39, 43, 47, 51, 55, 60, 65, 71, 77, 83, 88, 98, 105, 115, 122, 132, 142, 155, 164, 178, 191, 206, 220, 236, 252, 272, 290, 311, 332, 356, 378, 407, 434, 464

Offset

0,2

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

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 6th equation.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x) * (f(-x^5) / f(-x, -x^4))^3 in powers of x where f(,) is the Ramanujan general theta function.

Expansion of f(-x^2, -x^3) * G(x)^2 in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015

Euler transform of period 5 sequence [ 2, -1, -1, 2, -1, ...].

G.f.: (Sum_{k in Z} (-1)^k * x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(5*k + 1))^2.

a(n) = 3 * A053266(n) - A053262(n) unless n=0.

a(n) ~ sqrt(5+2*sqrt(5)) * exp(sqrt(2*n/15)*Pi)/ (5*sqrt(2*n)). - Vaclav Kotesovec, Dec 17 2016

Example

G.f. = 1 + 2*x + 2*x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 3*x^7 + 3*x^8 + ...
G.f. = 1/q + 2*q^119 + 2*q^239 + q^359 + 2*q^479 + 3*q^599 + 4*q^719 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -2, 1, 1, -2, 1}[[
Mod[k, 5, 1]]], {k, n}], {x, 0, n}];
nmax = 100; CoefficientList[Series[Product[(1-x^k)/((1 - x^(5*k-1))*(1 - x^(5*k-4)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 17 2016 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -2, 1, 1, -2][k%5+1]), n))};

Crossrefs

Cf. A053262, A053266.

Sequence in context: A352888 A354267 A366804 * A072549 A239481 A200114

Adjacent sequences: A259907 A259908 A259909 * A259911 A259912 A259913

Keyword

nonn

Author

Michael Somos, Jul 07 2015

Status

approved