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

Author

Michael Somos, Jul 07 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 + 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 + ...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 6th equation.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

Crossrefs

Cf. A053262, A053266.

Programs

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 *)

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))};

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