A259920 Expansion of phi(-x^5) * f(-x^5) / f(-x, -x^4) in powers of x where phi() and f() are Ramanujan theta functions.

1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 5, 3, 5, 2, 6, 3, 6, 3, 7, 4, 7, 5, 9, 5, 9, 5, 11, 6, 11, 7, 14, 7, 15, 9, 17, 9, 17, 9, 21, 11, 21, 12, 25, 13, 25, 15, 29, 16, 31, 17, 35, 19, 37, 21, 42, 22, 44, 25, 49, 27, 52, 29, 58, 32, 61

Offset: 0

Author

Michael Somos, Jul 08 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^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^-1 + q^59 + q^119 + q^179 + 2*q^239 + q^359 + q^419 + 2*q^479 + q^539 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 8th equation.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A053256, A053266.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 0, -1, 2, -1, 0, 0, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {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, 2, -1, 0, 0, -1][k%10+1]), n))};

Formula

Expansion of f(-x^5)^3 / (f(-x^10) * f(-x^2, -x^3)) in powers of x where f(,) is the Ramanujan general theta function.

Expansion of phi(-x^5) * G(x) 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 10 sequence [ 1, 0, 0, 1, -2, 1, 0, 0, 1, -1, ...].

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