A133738 Expansion of product of 3rd order mock theta function phi(q) and Ramanujan theta function f(-q) in powers of q.

1, 0, -2, -2, 2, 2, -2, 0, 2, 4, -2, -4, 2, 0, -2, -2, 2, 4, -4, -4, 2, 2, -2, 0, 4, 4, 0, -6, 2, 0, -2, 0, 2, 6, -4, -4, 4, 0, -4, -2, 0, 4, -2, -4, 2, 0, 0, 0, 4, 4, -2, -6, 2, 0, -6, 2, 2, 8, 0, -4, 2, 0, 0, 0, 2, 2, -6, -4, 2, 0, -2, 0, 4, 4, 0, -6, 2, -2

Offset: 0

Author

Michael Somos, Sep 22 2007

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*x^2 - 2*x^3 + 2*x^4 + 2*x^5 - 2*x^6 + 2*x^8 + 4*x^9 - 2*x^10 + ...
		

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. A010815, A053250.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, With[ {m = Quotient[ Sqrt[24 n + 1] - 1, 6]}, SeriesCoefficient[ Sum[ (-1)^k x^(k (3 k + 1)/2) (1 + x^k) / (1 + x^(2 k)), {k, -m, m}], {x, 0, n}]]]; (* Michael Somos, Jul 26 2015 *)

PARI

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, (sqrtint(24*n + 1) - 1) \ 6, (-1)^k * x^(k*(3*k + 1)/2) * (1 + x^k) / (1 + x^(2*k)), x * O(x^n)), n))};

PARI

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

Formula

G.f.: Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2) * (1 + x^k) / (1 + x^(2*k)).

G.f.: ( Product_{k>0} 1 - x^k ) * ( 1 + Sum_{k>0} x^k^2 / ((1 + x^2) * (1 + x^4) * ... * (1 + x^(2*k))) ).

Convolution of A053250 and A010815.