A215346 - OEIS

%I #21 Mar 12 2021 22:24:46

%S 1,-4,8,-16,34,-64,112,-192,319,-512,808,-1248,1886,-2816,4144,-6016,

%T 8643,-12288,17296,-24144,33442,-45952,62720,-85056,114620,-153600,

%U 204728,-271456,358204,-470528,615344,-801408,1039621,-1343488,1729920,-2219808,2838920

%N Expansion of (1/q) * phi(-q) * phi(q^4) / (phi(q) * psi(q^8)) in powers of q where phi(), psi() are Ramanujan theta functions.

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

%C A058516, A176143, A214035, A215346 are all essentially the same sequence. - _N. J. A. Sloane_, Aug 08 2012

%H G. C. Greubel, <a href="/A215346/b215346.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)^2 * eta(q^8)^3 / (eta(q^2)^3 * eta(q^16)^2))^2 in powers of q.

%F Euler transform of period 16 sequence [ -4, 2, -4, 2, -4, 2, -4, -4, -4, 2, -4, 2, -4, 2, -4, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A215348.

%F a(n) =  (-1)^n * A214035(n). a(2*n) = -4 * A131126(n). Convolution inverse of A215348.

%e 1/q - 4 + 8*q - 16*q^2 + 34*q^3 - 64*q^4 + 112*q^5 - 192*q^6 + 319*q^7 + ...

%t QP = QPochhammer; s = (QP[q]^2*(QP[q^8]^3/(QP[q^2]^3*QP[q^16]^2)))^2 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( eta(x + A)^2 * eta(x^8 + A)^3 / (eta(x^2 + A)^3 * eta(x^16 + A)^2))^2, n))}

%Y Cf. A000122, A000700, A010054, A121373, A131126, A214035, A215348.

%Y Cf. A058516, A176143, A214035, A215346.

%K sign

%O -1,2

%A _Michael Somos_, Aug 08 2012