A260149 Expansion of f(q) * phi(q) in powers of q where f() is a Ramanujan theta function and phi() is a 6th-order mock theta function.

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

Offset

0,3

Comments

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

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 4, 5th equation.

Links

Table of n, a(n) for n=0..79.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Convolution of A080995 and A053268.

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

Example

G.f. = 1 + 2*x^2 + 2*x^4 - 2*x^5 + 2*x^8 + 2*x^10 - 4*x^13 + 2*x^14 + ...

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 + 2 Sum[ x^(k (3 k + 1)/2) (1 + x^(2 k)) / (1 + x^(3 k)), {k, (Sqrt[ 24 n + 1] - 1) / 6}], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ -x, x] Sum[ (-1)^k x^k^2 QPochhammer[ x, x^2, k] / QPochhammer[ -x, x, 2 k], {k, 0, Sqrt @ n}], {x, 0, n}]];

Prog

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

Crossrefs

Cf. A080995, A053268.

Sequence in context: A373924 A028928 A343723 * A091379 A151758 A365538

Adjacent sequences: A260146 A260147 A260148 * A260150 A260151 A260152

Keyword

sign

Author

Michael Somos, Nov 08 2015

Status

approved