%I #12 Mar 12 2021 22:24:48

%S 1,1,1,1,2,0,1,1,2,1,2,1,3,2,2,2,3,1,3,2,5,3,5,2,6,3,6,3,7,4,7,5,9,5,

%T 9,5,11,6,11,7,14,7,15,9,17,9,17,9,21,11,21,12,25,13,25,15,29,16,31,

%U 17,35,19,37,21,42,22,44,25,49,27,52,29,58,32,61

%N Expansion of phi(-x^5) * f(-x^5) / f(-x, -x^4) in powers of x where phi() and f() are Ramanujan theta functions.

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

%C Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 8th equation.

%H G. C. Greubel, <a href="/A259920/b259920.txt">Table of n, a(n) for n = 0..1000</a>

%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 Expansion of f(-x^5)^3 / (f(-x^10) * f(-x^2, -x^3)) in powers of x where f(,) is the Ramanujan general theta function.

%F Expansion of phi(-x^5) * G(x) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function. - _Michael Somos_, Jul 09 2015

%F Euler transform of period 10 sequence [ 1, 0, 0, 1, -2, 1, 0, 0, 1, -1, ...].

%F G.f.: (Sum_{k in Z} (-1)^k * x^(5*k^2)) / (Product_{k in Z} 1 - x^abs(5*k + 1)).

%e G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + ...

%e G.f. = q^-1 + q^59 + q^119 + q^179 + 2*q^239 + q^359 + q^419 + 2*q^479 + q^539 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 0, -1, 2, -1, 0, 0, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^ [1, -1, 0, 0, -1, 2, -1, 0, 0, -1][k%10+1]), n))};

%Y Cf. A053256, A053266.

%K nonn

%O 0,5

%A _Michael Somos_, Jul 08 2015