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A113428 Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.

%I A113428 #36 Feb 16 2025 08:32:59
%S A113428 1,0,-1,-1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,
%T A113428 0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,
%U A113428 -1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0
%N A113428 Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.
%C A113428 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A113428 See the Hardy-Wright reference for the identity given as g.f. formula below. - _Wolfdieter Lang_, Oct 28 2016
%D A113428 G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
%D A113428 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 356, p. 284.
%H A113428 Seiichi Manyama, <a href="/A113428/b113428.txt">Table of n, a(n) for n = 0..10000</a>
%H A113428 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A113428 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A113428 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-Ramanujan Identities.html">Rogers-Ramanujan Identities</a>
%F A113428 Expansion of G(x) * f(-x) in powers of x where G() is the g.f. of A003114.
%F A113428 Euler transform of period 5 sequence [ 0, -1, -1, 0, -1, ...].
%F A113428 |a(n)| is the characteristic function of the numbers in A057569.
%F A113428 The exponents in the q-series q * A(q^40) are the square of the numbers in A090771.
%F A113428 G.f.: Sum_{k in Z} (-1)^k * x^((5*k^2 + k)/2) = Prod_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)).
%F A113428 Convolution of A003114 and A010815.
%F A113428 From _Wolfdieter Lang_, Oct 30 2016: (Start)
%F A113428 a(n) = (-1)^k if n = b(2*k+1) for k >= 0, a(n) = (-1)^k if n = b(2*k), for k >= 1, and a(n) = 0 otherwise, where b(n) = A057569(n). See the third formula.
%F A113428 G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+1)/2)*(1-x^(2*(2*n+1))). See the Hardy reference, p. 93, eq. (6.11.1) with k=2, a=x and C_n = 1.
%F A113428 (End)
%F A113428 G.f.: Sum_{n >= 0} x^(n^2)*Product_{k >= n+1} 1 - x^k. Cf. A113429. - _Peter Bala_, Feb 12 2021
%e A113428 G.f. = 1 - x^2 - x^3 + x^9 + x^11 - x^21 - x^24 + x^38 + x^42 - x^60 - x^65 + ...
%e A113428 G.f. = q - q^81 - q^121 + q^361 + q^441 - q^841 - q^961 + q^1521 + q^1681 + ...
%t A113428 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5], {x, 0, n}]; (* _Michael Somos_, Jan 06 2016 *)
%o A113428 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k * [1, 0, 1, 1, 0][k%5 + 1], 1 + x * O(x^n)), n))};
%Y A113428 Cf. A003114, A057569, A090771, A010815, A113429.
%K A113428 sign,easy
%O A113428 0,1
%A A113428 _Michael Somos_, Oct 31 2005