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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
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 (list; graph; refs; listen; history; text; internal format)
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 2, 2nd equation.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: A216265 A130277 A109135 * A274850 A215594 A230291

Adjacent sequences:  A264133 A264134 A264135 * A264137 A264138 A264139

KEYWORD

sign

AUTHOR

Michael Somos, Nov 03 2015

STATUS

approved

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Last modified July 5 00:01 EDT 2020. Contains 335457 sequences. (Running on oeis4.)