A214157 - OEIS

%I #14 Mar 12 2021 22:24:46

%S 1,1,0,1,2,0,-1,0,-1,-1,0,-2,0,4,1,-2,3,4,-2,-3,-1,-2,-2,-2,-5,0,9,3,

%T -4,8,12,-4,-7,-1,-6,-6,-4,-12,-1,22,6,-10,17,24,-9,-16,-3,-12,-11,-8,

%U -25,-1,45,14,-20,36,52,-18,-32,-6,-25,-24,-16,-50,-2,88

%N Expansion of (1/x) * (f(-x^2, -x^11) * f(-x^5, -x^8) * f(-x^6, -x^7)) / (f(-x, -x^12) * f(-x^3, -x^10) * f(-x^4, -x^9)) in powers of x where f(, ) is Ramanujan's general theta function.

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

%H G. C. Greubel, <a href="/A214157/b214157.txt">Table of n, a(n) for n = -1..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 Euler transform of period 13 sequence [1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v + u*v^3 - u^3*v^2 + 2*u*v * (1 + u - v + u*v).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^3 - u) * (u^3 - v) - 3*u*v * (1 + u + v) * (u*v - u - v).

%F G.f.: (1/x) * Product_{k>0} (1 - x^k)^-Kronecker(13, k).

%F a(n) = A092876(n) + A133099(n) unless n=0.

%F Convolution inverse of A092876.

%e G.f. = 1/x + 1 + x^2 + 2*x^3 - x^5 - x^7 - x^8 - 2*x^10 + 4*x^12 + x^13 - 2*x^14 + ...

%t a[ n_] := SeriesCoefficient[ 1/x Product[ (1 - x^k) ^ -KroneckerSymbol[ 13, k], {k, n+1}], {x, 0, n}];

%o (PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod( k=1, n, (1 - x^k)^-kronecker( 13, k), 1 + x * O(x^n)), n))};

%Y Cf. A092876, A133099.

%K sign

%O -1,5

%A _Michael Somos_, Jul 05 2012