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Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.
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%I #55 Feb 17 2021 09:57:50

%S 1,-1,0,0,-1,0,0,1,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,

%T 0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0

%N Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.

%C For the g.f. identity see the Hardy-Wright reference, Theorem 355 on p. 284. - _Wolfdieter Lang_, Oct 28 2016

%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

%H Seiichi Manyama, <a href="/A113429/b113429.txt">Table of n, a(n) for n = 0..10000</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>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-Ramanujan Identities.html">Rogers-Ramanujan Identities</a>

%F Euler transform of period 5 sequence [-1, 0, 0, -1, -1, ...].

%F |a(n)| is the characteristic function of A085787.

%F G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k-1)) * (1 - x^(5*k-4)) = Sum_{k in Z} (-1)^k * x^((5*k^2+3*k)/2).

%F f(a, b) = Sum_{k in Z} a^((k^2+k)/2) * b^((k^2-k)/2) is Ramanujan's general theta function.

%F G.f.: Sum_{n>=0} (x^(n*(n+1)) * Product_{k>=n+1} (1-x^k)). - _Joerg Arndt_, Apr 07 2011

%F From _Wolfdieter Lang_, Oct 30 2016: (Start)

%F a(n) = (-1)^k if n = b(2*k) for k >= 0, a(n) = (-1)^k if n = b(2*k-1), for k >= 1, and a(n) = 0 otherwise, where b(n) = A085787(n). See the second formula.

%F G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+3)/2)*(1-x^(2*n+1)). See the Hardy reference, p. 93, G_1(x,x) from eq. (6.11.1) with C_n(x,x) = 1.

%F (End)

%F G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n-3)/2)*(1-x^(4*(2*n+1)). Reordered G_1(x,x) from the preceding formula. This is G_4(x,x) from Hardy, p. 93, eq. (6.11.1) with C_n(x,x) = 1. Note that Hardy uses only G_0, G_1 and G_2. - _Wolfdieter Lang_, Nov 01 2016

%F a(n) = -(1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%e G.f. = 1 - x - x^4 + x^7 + x^13 - x^18 - x^27 + x^34 + x^46 - x^55 - x^70 + ...

%e G.f. = q^9 - q^49 - q^169 + q^289 + q^529 - q^729 - q^1089 + q^1369 + q^1849 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^5] QPochhammer[ x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* _Michael Somos_, Jun 26 2017 *)

%t a[ n_] := Module[{m = 40 n + 9, k}, If[IntegerQ[k = Sqrt[m]], If[Mod[k, 10] == 7, k = -k]; (-1)^Quotient[k, 10], 0]]; (* _Michael Somos_, Jun 26 2017 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k*[1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};

%o (PARI) {a(n) = my(m, k); if( n<0, 0, issquare(m = 40*n + 9, &k), if( k%10==7, k=-k); (-1)^(k\10), 0)}; /* _Michael Somos_, Oct 29 2016 */

%Y Cf. A085787, A113428.

%K sign,easy

%O 0,1

%A _Michael Somos_, Oct 31 2005