A132970 Expansion of phi(-x) * chi(-x) in powers of x 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, -1220, 1331

Offset

0,2

Comments

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

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 60, Eqs. (26.64),(26.65),(26.66)

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

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)^3 / eta(q^2)^2 in powers of q.

Euler transform of period 2 sequence [ -3, -1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 48^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A085140.

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

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

a(n) = (-1)^n * A132969(n). a(n) = A124226(n) unless n=1.

a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Oct 14 2017

Example

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 + ...
G.f. = 1/q - 3*q^23 + 2*q^47 - q^71 + 5*q^95 - 5*q^119 + 3*q^143 - 5*q^167 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jul 20 2015 *)

Prog

(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 * (-1)^k * x^k^2, 1), n))};
(PARI) {a(n) = my(A) ; if( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^3 / eta(x^2 + A)^2, n))};

Crossrefs

CF. A124226, A132969.

Sequence in context: A091595 A246837 A132969 * A192022 A208608 A209577

Adjacent sequences: A132967 A132968 A132969 * A132971 A132972 A132973

Keyword

sign

Author

Michael Somos, Sep 04 2007

Status

approved