cp's OEIS Frontend

A215600 Expansion of psi(-x)^2 * f(-x)^6 in powers of x where psi(), f() are Ramanujan theta functions.

%I A215600 #16 Feb 16 2025 08:33:18
%S A215600 1,-8,22,-16,-27,40,-18,80,-94,-40,0,-48,359,-80,-130,-320,0,160,214,
%T A215600 400,-230,-152,-594,416,-343,240,518,-400,0,200,830,-592,-396,-776,0,
%U A215600 -400,1098,200,0,1120,729,-552,-2068,272,-1670,800,0,400,594,1480,598,48
%N A215600 Expansion of psi(-x)^2 * f(-x)^6 in powers of x where psi(), f() are Ramanujan theta functions.
%C A215600 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A215600 G. C. Greubel, <a href="/A215600/b215600.txt">Table of n, a(n) for n = 0..1000</a>
%H A215600 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A215600 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A215600 Expansion of psi(-x)^8 * chi(-x^2)^6 = f(-x)^8 / chi(-x^2)^2 in powers of x where psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Sep 03 2013
%F A215600 Expansion of q^(-1/2) * (eta(q)^4 * eta(q^4) / eta(q^2))^2 in powers of q.
%F A215600 Euler transform of period 4 sequence [ -8, -6, -8, -8, ...].
%F A215600 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^15 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A225564. - _Michael Somos_, Sep 03 2013
%F A215600 Convolution square of A215597.
%F A215600 a(2*n) = A215601(n). - _Michael Somos_, Sep 03 2013
%e A215600 1 - 8*x + 22*x^2 - 16*x^3 - 27*x^4 + 40*x^5 - 18*x^6 + 80*x^7 - 94*x^8 + ...
%e A215600 q - 8*q^3 + 22*q^5 - 16*q^7 - 27*q^9 + 40*q^11 - 18*q^13 + 80*q^15 - 94*q^17 + ...
%t A215600 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^6 EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (2 q^(1/4)), {q, 0, n}] (* _Michael Somos_, Sep 03 2013 *)
%t A215600 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^8 / QPochhammer[ q^2, q^4]^2, {q, 0, n}] (* _Michael Somos_, Sep 03 2013 *)
%o A215600 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^4 * eta(x^4 + A) / eta(x^2 + A))^2, n))}
%Y A215600 Cf. A215597, A215601, A225564.
%K A215600 sign
%O A215600 0,2
%A A215600 _Michael Somos_, Aug 16 2012