A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

1, 3, 2, 1, 5, 5, 3, 5, 6, 10, 10, 8, 13, 15, 15, 16, 23, 27, 25, 30, 35, 40, 42, 45, 55, 66, 68, 70, 86, 95, 100, 110, 125, 141, 150, 161, 185, 207, 215, 235, 266, 293, 310, 335, 375, 410, 438, 470, 521, 575, 610, 653, 725, 785, 835, 900, 983, 1070, 1140

Offset: 0

Author

Michael Somos, Sep 04 2007

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 3*x + 2*x^2 + x^3 + 5*x^4 + 5*x^5 + 3*x^6 + 5*x^7 + 6*x^8 + 10*x^9 + ...
G.f. = 1/q + 3*q^23 + 2*q^47 + q^71 + 5*q^95 + 5*q^119 + 3*q^143 + 5*q^167 +...
		

References

• N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; top of p. 60.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A053250, A053251, A124226, A132970.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 - x^(2*k)) * ( (1 + x^k) / (1 + x^(2*k)) )^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + x*O(x^n)) * sum(k=1, sqrtint(n), 2 * x^k^2, 1), n))};

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A) * eta(x^4 + A))^3, n))};

Formula

Expansion of phi(q) + 2 * psi(q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.

Expansion of q^(1/24) * eta(q^2)^7 / (eta(q) * eta(q^4))^3 in powers of q.

Euler transform of period 4 sequence [ 3, -4, 3, -1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t).

G.f.: ( Sum_{k in Z} x^k^2 ) * ( Product_{k>0} (1 + x^(2*k-1)) ).

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

a(n) = (-1)^n * A132970(n). a(n) = (-1)^n * A124226(n) unless n=1.

a(n) ~ exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Sep 08 2015