A137676 - OEIS

%I #15 Mar 12 2021 22:24:45

%S 1,1,0,0,1,1,0,0,1,2,0,0,2,2,0,0,3,3,0,0,4,4,0,0,5,6,0,0,7,7,0,0,9,10,

%T 0,0,12,12,0,0,15,16,0,0,19,20,0,0,24,26,0,0,30,31,0,0,37,40,0,0,46,

%U 48,0,0,57,60,0,0,69,72,0,0,84,89,0,0,102,106,0

%N Expansion of f(-x^2, -x^3) / f(-x, -x^3) in powers of x where f(, ) is Ramanujan's general 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="/A137676/b137676.txt">Table of n, a(n) for n = 0..1000</a>

%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/cbms/066">q-series</a>, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 36, Eq. (4.11). MR0858826 (88b:11063).

%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 f(-x^2) * f(-x^5) / (f(-x^4) * f(-x, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

%F Expansion of (f(-x^13, -x^17) + x * f(-x^7, -x^23)) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.

%F Euler transform of period 20 sequence [ 1, -1, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, ...].

%F G.f.: Sum_{k>=0} x^k^2 / (Product_{j=1..k} 1 - x^(4*j)).

%F a(4*n) = A122129(n). a(4*n + 1) = A122135(n). a(4*n + 2) = a(4*n + 3) = 0.

%F G.f.: (Sum_{k in Z} (-1)^k^2 * x^(k * (5*k + 1) / 2)) / (Sum_{k in Z} (-1)^k^2 * x^(k * (2*k + 1))). - _Michael Somos_, Oct 08 2015

%e G.f. = 1 + x + x^4 + x^5 + x^8 + 2*x^9 + 2*x^12 + 2*x^13 + 3*x^16 + 3*x^17 + ...

%e G.f. = 1/q + q^9 + q^39 + q^49 + q^79 + 2*q^89 + 2*q^119 + 2*q^129 + 3*q^159 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / (QPochhammer[ x^4] QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* _Michael Somos_, Oct 08 2015 *)

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2) / QPochhammer[ x^4, x^4, k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* _Michael Somos_, Oct 08 2015 *)

%t a[ n_] := SeriesCoefficient[ Sqrt[2] x^(1/8) QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5] / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}]; (* _Michael Somos_, Oct 08 2015 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, k, 1 - x^(4*i), 1 + x * O(x^(n - k^2)))), n))};

%Y Cf. A122129, A122135.

%K nonn

%O 0,10

%A _Michael Somos_, Feb 04 2008