A224877 - OEIS

%I #22 Mar 12 2021 22:24:46

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

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

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

%N Expansion of f(x^29, x^31) - x^3 * f(x^11, x^49) in powers of x where f() is a Ramanujan theta function.

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

%C This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^17, b = x^3.

%H G. C. Greubel, <a href="/A224877/b224877.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>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>

%F Expansion of f(-x^3, -x^7) * f(x^7, x^13) / f(-x^10) = f(-x^10) * f(-x^6, -x^14) / f(x^3, x^17) in powers of x where f() is a Ramanujan theta function.

%F Euler transform of period 40 sequence [ 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, ...].

%F G.f.: Sum_{k in Z} x^(k * (30*k + 10)) * (x^(-9*k) - x^(9*k + 3)).

%F a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = a(11*n + 2) = a(11*n + 4) = a(11*n + 5) = a(11*n + 6) = a(11*n + 10) = 0.

%e G.f. = 1 - x^3 - x^14 + x^29 + x^31 - x^52 - x^85 + x^118 + x^122 - x^161 - x^216 + ...

%e G.f. = q - q^361 - q^1681 + q^3481 + q^3721 - q^6241 - q^10201 + q^14161 + q^14641 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^10] QPochhammer[ x^7, x^10] QPochhammer[ -x^7, x^20] QPochhammer[ -x^13, x^20] QPochhammer[ x^20], {x, 0, n}];

%o (PARI) {a(n) = local(m); if( n<0 || !issquare( 120*n + 1, &m), 0, if( m%20 > 9, m = -m); if( m%20 != 1, 0, 1 - (m\20)%3))};

%K sign

%O 0,1

%A _Michael Somos_, Jul 23 2013