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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.
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%I #14 Feb 01 2021 03:09:19

%S 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,

%T 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,

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

%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.

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

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 4, 5th equation.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Convolution of A080995 and A053268.

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

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

%t 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}]];

%t 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}]];

%o (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))};

%Y Cf. A080995, A053268.

%K sign

%O 0,3

%A _Michael Somos_, Nov 08 2015