A261369 - OEIS

%I #17 Mar 12 2021 22:24:48

%S 1,-2,5,-12,24,-46,86,-152,262,-442,725,-1168,1852,-2886,4436,-6736,

%T 10103,-14994,22040,-32092,46336,-66380,94378,-133256,186926,-260576,

%U 361126,-497716,682340,-930774,1263624,-1707672,2297737,-3078850,4109022,-5462924,7236280

%N Expansion of (psi(-x^3) / f(x))^2 in powers of x where psi(), f() are Ramanujan theta functions.

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

%H Vaclav Kotesovec, <a href="/A261369/b261369.txt">Table of n, a(n) for n = 0..2000</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 q^(-2/3) * (eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2)^3 * eta(q^6)))^2 in powers of q.

%F Euler transform of period 12 sequence [ -2, 4, -4, 2, -2, 4, -2, 2, -4, 4, -2, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A262930. - _Michael Somos_, Nov 07 2015

%F a(n) = A187153(3*n + 2) = A213265(3*n + 2) = A262930(3*n + 2). - _Michael Somos_, Nov 07 2015

%F Convolution square of A139135. - _Michael Somos_, Nov 07 2015

%F a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (8*3^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017

%e G.f. = 1 - 2*x + 5*x^2 - 12*x^3 + 24*x^4 - 46*x^5 + 86*x^6 - 152*x^7 + ...

%e G.f. = q^2 - 2*q^5 + 5*q^8 - 12*q^11 + 24*q^14 - 46*q^17 + 86*q^20 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(3/2)]^2 / (2 x^(3/4) QPochhammer[ -x]^2), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^3 * eta(x^6 + A)))^2, n))};

%Y Cf. A139135, A187153, A213265, A262930.

%K sign

%O 0,2

%A _Michael Somos_, Aug 16 2015