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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

%I

%S 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,

%T -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,

%U 6,0,0,-4,-2,0,8,-4,0,-2,2,0,-2,-4,-2,-4,4,0,8,2

%N 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.

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 2, 2nd equation.

%H Vaclav Kotesovec, <a href="/A264136/b264136.txt">Table of n, a(n) for n = 0..1000</a> (corrected previous b-file from G. C. Greubel)

%F Convolution of A010815 and A053268.

%F 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)).

%e 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 + ...

%t 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}]];

%t 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 *)

%o (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))};

%Y Cf. A010815, A053268.

%K sign

%O 0,2

%A _Michael Somos_, Nov 03 2015

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Last modified August 8 09:16 EDT 2020. Contains 336293 sequences. (Running on oeis4.)