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A259358 Expansion of f(-x^5)^2 / f(-x^2, -x^3) in powers of x where f(,) is the Ramanujan general theta function.

%I A259358 #16 Feb 16 2025 08:33:26
%S A259358 1,0,1,1,1,0,2,1,2,2,2,2,3,2,4,4,5,4,6,5,7,7,8,8,11,10,12,13,15,14,18,
%T A259358 17,21,21,24,25,29,29,34,35,39,40,47,47,53,55,61,63,72,73,82,86,94,97,
%U A259358 109,112,124,129,141,147,162,167,183,192,208,217,237
%N A259358 Expansion of f(-x^5)^2 / f(-x^2, -x^3) in powers of x where f(,) is the Ramanujan general theta function.
%C A259358 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259358 Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
%D A259358 G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, 2012, see p. 12, Entry 2.1.3, Equation (2.1.22).
%D A259358 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 4.
%H A259358 G. C. Greubel, <a href="/A259358/b259358.txt">Table of n, a(n) for n = 0..2500</a>
%H A259358 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259358 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259358 Expansion of f(-x^5) * f(-x, -x^4) / f(-x) in powers of x where f(,) is the Ramanujan general theta function.
%F A259358 Expansion of f(-x^5) * H(x) in powers of x where f() is a Ramanujan theta funcation and H() is a Rogers-Ramanujan function. - _Michael Somos_, Jul 09 2015
%F A259358 Euler transform of period 5 sequence [ 0, 1, 1, 0, -1, ...].
%F A259358 G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-3)) * (1 - x^(5*k-2))).
%F A259358 Convolution of A035959 and A113429.
%e A259358 G.f. = 1 + x^2 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + ...
%e A259358 G.f. = q^47 + q^287 + q^407 + q^527 + 2*q^767 + q^887 + 2*q^1007 + ...
%t A259358 a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}];
%o A259358 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -1, -1, 0][k%5+1]), n))};
%Y A259358 Cf. A035959, A113429, A259357.
%K A259358 nonn
%O A259358 0,7
%A A259358 _Michael Somos_, Jun 24 2015