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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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 2, 2nd equation.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
Sequence in context: A336694 A130277 A109135 * A274850 A349437 A215594
KEYWORD
sign
AUTHOR
Michael Somos, Nov 03 2015
STATUS
approved

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Last modified July 13 02:50 EDT 2024. Contains 374265 sequences. (Running on oeis4.)