cp's OEIS Frontend

A230322 Expansion of f(-x^3, -x^4) / f(-x^2,-x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.

%I A230322 #15 Feb 16 2025 08:33:20
%S A230322 1,0,1,-1,0,0,0,1,-1,1,-1,0,0,-1,2,-1,2,-2,0,0,-1,3,-3,3,-3,1,0,-2,5,
%T A230322 -4,5,-5,1,0,-3,7,-7,8,-7,2,0,-5,11,-10,12,-11,3,1,-7,15,-16,17,-15,5,
%U A230322 1,-11,22,-22,25,-22,7,2,-15,31,-33,35,-30,11,2,-22
%N A230322 Expansion of f(-x^3, -x^4) / f(-x^2,-x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.
%C A230322 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A230322 G. C. Greubel, <a href="/A230322/b230322.txt">Table of n, a(n) for n = 0..1000</a>
%H A230322 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A230322 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A230322 Euler transform of period 7 sequence [ 0, 1, -1, -1, 1, 0, 0, ...].
%F A230322 G.f.: Product_{k>0} (1 - x^(7*k - 3)) * (1 - x^(7*k - 4)) / ((1 - x^(7*k - 2)) * (1 - x^(7*k - 5))).
%F A230322 - a(n) = A229894(7*n + 1).
%F A230322 G.f.: B(x) / C(x), where B(x) is the g.f. of A375150 and C(x) is the g.f. of A375107. - _Seiichi Manyama_, Aug 03 2024
%e A230322 G.f. = 1 + x^2 - x^3 + x^7 - x^8 + x^9 - x^10 - x^13 + 2*x^14 - x^15 + ...
%e A230322 G.f. = 1/q + q^13 - q^20 + q^48 - q^55 + q^62 - q^69 - q^90 + 2*q^97 - ...
%t A230322 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^7] QPochhammer[ x^4, x^7] / (QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7]), {x, 0, n}]
%o A230322 (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 0, -1, 1, 1, -1, 0][k%7+1], 1 + x * O(x^n)), n))}
%Y A230322 Cf. A229894, A375107, A375150.
%K A230322 sign
%O A230322 0,15
%A A230322 _Michael Somos_, Oct 16 2013