A123655 - OEIS

%I #32 Mar 12 2021 22:24:44

%S 1,2,4,8,14,24,40,64,101,156,236,352,518,752,1080,1536,2162,3018,4180,

%T 5744,7840,10632,14328,19200,25591,33932,44776,58816,76918,100176,

%U 129952,167936,216240,277476,354864,452392,574958,728568,920600,1160064

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

%C Ramanujan theta functions: phi(q) (A000122), psi(q) (A010054).

%C Number 12 of the 14 eta-quotients listed in Table 2 of Moy 2013. - _Michael Somos_, Sep 19 2013

%H Seiichi Manyama, <a href="/A123655/b123655.txt">Table of n, a(n) for n = 1..1000</a>

%H Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

%H Richard Moy, <a href="http://arxiv.org/abs/1309.4320">Congruences among power series coefficients of modular forms</a>, arXiv:1309.4320

%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^16)^2 / (eta(q)^2 * eta(q^8)) in powers of q.

%F Euler transform of period 16 sequence [ 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v * (1 + 4*u) * (1 + 2*v).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/8 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A185338.

%F a(n) is odd iff n is an odd square. If n>2 is a power of 2 then the highest power of 2 dividing a(n) is (n/2)^3. - _Michael Somos_, Feb 18 2007

%F 4 * a(n) = A007096(n) unless n=0. -(-1)^n * a(n) = A208605(n). Convolution inverse of A185338.

%F G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k)) * (1 + x^(8*k))^2. _Michael Somos_, Sep 19 2013

%F a(2*n) = 2 * A107035(n). a(2*n + 1) = A093160(n). - _Michael Somos_, Sep 19 2013

%F a(n) ~ exp(sqrt(n)*Pi) / (2^(9/2) * n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017

%e G.f. = q + 2*q^2 + 4*q^3 + 8*q^4 + 14*q^5 + 24*q^6 + 40*q^7 + 64*q^8 + 101*q^9 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^4] / EllipticTheta[ 4, 0, q] / 2, {q, 0, n}]; (* _Michael Somos_, Sep 19 2013 *)

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

%Y Cf. A007096, A185338, A093160, A107035, A185338.

%K nonn

%O 1,2

%A _Michael Somos_, Oct 04 2006