cp's OEIS Frontend

A137828 Expansion of phi(x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.

%I A137828 #13 Feb 16 2025 08:33:07
%S A137828 1,2,0,0,4,4,0,0,9,12,0,0,20,24,0,0,42,50,0,0,80,92,0,0,147,172,0,0,
%T A137828 260,296,0,0,445,510,0,0,744,840,0,0,1215,1372,0,0,1944,2176,0,0,3059,
%U A137828 3424,0,0,4740,5268,0,0,7239,8040,0,0,10920,12072,0,0,16286
%N A137828 Expansion of phi(x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%C A137828 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A137828 G. C. Greubel, <a href="/A137828/b137828.txt">Table of n, a(n) for n = 0..1000</a>
%H A137828 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A137828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A137828 Expansion of q^(1/3) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^4) in powers of q.
%F A137828 Euler transform of period 4 sequence [ 2, -3, 2, 1, ...].
%F A137828 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 6^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A051136.
%F A137828 G.f.: Product_{k>0} (1 + x^k) / ( (1 - x^k) * (1 + x^(2*k))^4 ).
%F A137828 a(4*n + 2) = a(4*n + 3) = 0.
%F A137828 a(4*n) = A051136(n). a(4*n + 1) = 2 * A137829(n).
%e A137828 G.f. = 1 + 2*x + 4*x^4 + 4*x^5 + 9*x^8 + 12*x^9 + 20*x^12 + 24*x^13 + 42*x^16 + ...
%e A137828 G.f. = 1/q + 2*q^2 + 4*q^11 + 4*q^14 + 9*q^23 + 12*q^26 + 20*q^35 + 24*q^38 + ...
%t A137828 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* _Michael Somos_, Oct 04 2015 *)
%o A137828 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x^4 + A)^4 / eta(x + A)^2, n))};
%Y A137828 Cf. A051136, A137829.
%K A137828 nonn
%O A137828 0,2
%A A137828 _Michael Somos_, Feb 12 2008