A264136 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, -2, 2, -2, 0, -2, 4, 0, 2, -2, 2, -4, -2, 0, 6, -2, 0, -4, 4, 0, -2, -2, 2, -4, 2, 2, 8, -2, -2, -4, 2, 0, 2, -2, 0, -4, -2, 0, 8, -2, 0, -4, 6, 0, -2, 0, 0, -4, 0, -2, 6, -2, -2, -4, 4, 2, 6, 0, 0, -4, -2, 0, 8, -4, 0, -2, 2, 0, -2, -4, -2, -4, 4, 0, 8, 2

Offset

0,2

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 2, 2nd equation.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (corrected previous b-file from G. C. Greubel)

Formula

Convolution of A010815 and A053268.

G.f.: Sum_{k in Z} x^(6*k^2 + k) / (1 - x^k + x^(2*k)) - 2 * Sum_{k in Z} x^(6*k^2 - 2*k) / (1 + x^(3*k - 1)).

Example

G.f. = 1 - 2*x + 2*x^2 - 2*x^3 - 2*x^5 + 4*x^6 + 2*x^8 - 2*x^9 + 2*x^10 - 4*x^11 + ...

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x] Sum[ (-1)^k x^k^2 QPochhammer[ x, x^2, k] / QPochhammer[ -x, x, 2*k], {k, 0, Sqrt@n}], {x, 0, n}]];
nmax = 122; CoefficientList[Series[QPochhammer[q]*Sum[(-1)^n*q^n^2*Product[1 - q^k, {k, 1, 2*n - 1, 2}] / Product[1 + q^k, {k, 1, 2*n}], {n, 0, Floor[Sqrt[nmax]]}], {q, 0, nmax}], q] (* G. C. Greubel, Mar 18 2018, fixed by Vaclav Kotesovec, Jun 15 2019 *)

Prog

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

Crossrefs

Cf. A010815, A053268.

Sequence in context: A336694 A130277 A109135 * A274850 A349437 A215594

Adjacent sequences: A264133 A264134 A264135 * A264137 A264138 A264139

Keyword

sign

Author

Michael Somos, Nov 03 2015

Status

approved