A190611 - OEIS

%I #13 Mar 12 2021 22:24:46

%S 1,-1,1,-1,1,-2,1,-2,1,-1,2,-2,1,0,2,-2,1,0,1,0,2,-2,2,0,1,-3,0,-1,2,

%T -2,2,-2,1,-2,0,-4,1,0,0,0,2,0,2,0,2,-2,0,0,1,-3,3,0,0,-2,1,-4,2,0,2,

%U -2,2,0,2,-2,1,0,2,0,0,0,4,0,1,-2,0,-3,0,-4,0

%N Expansion of f(q^3) * f(-q^8) * chi(-q^12) / chi(q) in powers of q where f(), chi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A190611/b190611.txt">Table of n, a(n) for n = 0..5000</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 Expansion of eta(q) * eta(q^4) * eta(q^6)^3 * eta(q^8) / (eta(q^2)^2 * eta(q^3) * eta(q^24)) in powers of q.

%F Euler transform of period 24 sequence [ -1, 1, 0, 0, -1, -1, -1, -1, 0, 1, -1, -2, -1, 1, 0, -1, -1, -1, -1, 0, 0, 1, -1, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 96^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129402.

%F a(n) = (-1)^n * A000377(n). a(24*n + 13) = a(24*n + 17) = a(24*n + 19) = a(24*n + 23) = 0.

%F a(2*n) = A000377(n). a(2*n + 1) = - A129402(n). a(3*n) = a(n). - _Michael Somos_, Nov 11 2015

%e G.f. = 1 - q + q^2 - q^3 + q^4 - 2*q^5 + q^6 - 2*q^7 + q^8 - q^9 + 2*q^10 + ...

%t a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n DivisorSum[ n, KroneckerSymbol[ -6, #] &]]; (* _Michael Somos_, Nov 11 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q] QPochhammer[ -q^3] QPochhammer[ q^8] QPochhammer[ q^12, -q^12], {q, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv( n, d, kronecker( -6, d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 * eta(x^8 + A) / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^24 + A)), n))};

%Y Cf. A000377, A129402.

%K sign

%O 0,6

%A _Michael Somos_, May 14 2011