A145726 - OEIS

%I #14 Mar 12 2021 22:24:45

%S 1,-1,0,0,-1,1,0,-1,0,1,0,0,1,-2,1,2,-3,1,1,-2,3,0,-3,1,2,-2,0,2,-6,3,

%T 4,-7,3,2,-5,6,2,-8,3,5,-6,2,4,-12,7,10,-15,6,5,-13,12,4,-18,7,11,-14,

%U 6,10,-24,14,20,-32,12,12,-29,24,9,-36,15,22,-30,13,22,-50,27,36,-63,26,24,-56,45,22,-69,30,42,-62,27

%N Expansion of q * psi(-q) * psi(-q^15) / (psi(-q^3) * psi(-q^5)) in powers of q where psi() is a Ramanujan 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="/A145726/b145726.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 Expansion of eta(q) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60) / (eta(q^2) * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30)) in powers of q.

%F Euler transform of a period 60 sequence.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).

%F G.f.: x * Product_{k>0} P(15, x^k) * P(60, x^k) where P(n, x) is the n-th cyclotomic polynomial.

%F a(n) = A145727(n) unless n=0. a(n) = -(-1)^n * A131794(n). a(2*n) = - A094022(n). Convolution inverse of A145725.

%e q - q^2 - q^5 + q^6 - q^8 + q^10 + q^13 - 2*q^14 + q^15 + 2*q^16 - 3*q^17 + ...

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A)), n))}

%Y Cf. A094022, A131794, A145725, A145727.

%K sign

%O 1,14

%A _Michael Somos_, Oct 23 2008