This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A215594 #17 Jun 10 2025 09:00:45
%S A215594 1,-2,2,-2,0,2,-4,6,-4,0,6,-12,14,-10,0,14,-26,30,-22,0,28,-52,60,-42,
%T A215594 0,54,-100,112,-78,0,100,-180,202,-140,0,174,-314,350,-240,0,296,-532,
%U A215594 588,-402,0,492,-876,966,-658,0,794,-1412,1550,-1050,0,1260,-2232
%N A215594 Expansion of f(-x, -x^4) / f(x, x^4) in powers of x where f(,) is Ramanujan's two-variable theta function.
%C A215594 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A215594 Let A(q) denote the g.f. of this sequence. Let m be a nonzero integer. The simple continued fraction expansions of the real numbers A(1/(2*m)) and A(1/(2*m+1)) may be predictable. For a given positive integer n, the sequence of the n-th partial denominators of the continued fractions are conjecturally polynomial or quasi-polynomial in m for m sufficiently large. An example is given below. Cf. A080054 and A098151. - _Peter Bala_, Jun 10 2025
%H A215594 G. C. Greubel, <a href="/A215594/b215594.txt">Table of n, a(n) for n = 0..1000</a>
%H A215594 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A215594 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A215594 Euler transform of period 10 sequence [ -2, 1, 0, -2, 0, -2, 0, 1, -2, 0, ...].
%F A215594 a(5*n + 4) = 0.
%e A215594 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^5 - 4*x^6 + 6*x^7 - 4*x^8 + 6*x^10 - 12*x^11 + ...
%e A215594 From _Peter Bala_, Jun 10 2025: (Start)
%e A215594 G.f.: A(q) = f(-q, -q^4) / f(q, q^4).
%e A215594 Simple continued fraction expansions of A(1/(2*m)):
%e A215594 m = 2 [0; 1 1 2 7 1 1 1 10 1 2 12 82 1 6 1 48 ...]
%e A215594 m = 3 [0; 1 2 2 17 1 1 2 14 2 2 24 318 1 1 1 1 ...]
%e A215594 m = 4 [0; 1 3 2 31 1 1 3 18 3 2 40 810 2 7 2 161 ...]
%e A215594 m = 5 [0; 1 4 2 49 1 1 4 22 4 2 60 1654 2 1 1 1 ...]
%e A215594 m = 6 [0; 1 5 2 71 1 1 5 26 5 2 84 2946 3 7 1 1 ...]
%e A215594 m = 7 [0; 1 6 2 97 1 1 6 30 6 2 112 4782 3 1 1 1 ...]
%e A215594 m = 8 [0; 1 7 2 127 1 1 7 34 7 2 144 7258 4 7 1 2 ...]
%e A215594 m = 9 [0; 1 8 2 161 1 1 8 38 8 2 180 10470 4 1 1 1 ...]
%e A215594 m = 10 [0; 1 9 2 199 1 1 9 42 9 2 220 14514 5 7 1 3 ...]
%e A215594 m = 11 [0; 1 10 2 241 1 1 10 46 10 2 264 19486 5 1 1 1 ...]
%e A215594 m = 12 [0; 1 11 2 287 1 1 11 50 11 2 312 25482 6 7 1 4 ...]
%e A215594 ...
%e A215594 The sequence of the 4th partial denominators [7, 17, 31, 49, ...] appears to be given by the polynomial 2*m^2 - 1 for m >= 2.
%e A215594 The sequence of the 11th partial denominators [12, 24, 40, 60, ...] appears to be given by the polynomial 2*(m^2 + m) for m >= 2.
%e A215594 The sequence of the 12th partial denominators [82, 318, 810, 1654, ...] appears to be given by the polynomial 2*(8*m^3 - 8*m^2 + 6*m - 3) for m >= 2.
%e A215594 The sequence of the 16th partial denominators appears to become quasi-polynomial in m for m >= 5. (End)
%t A215594 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A215594[n_] := SeriesCoefficient[f[-x, -x^4]/f[x, x^4], {x, 0, n}]; Table[A215594[n], {n,0,50}] (* _G. C. Greubel_, Jun 18 2017 *)
%o A215594 (PARI) {a(n) = local(A, s); if( n<0, 0, A = x * O(x^n); s = sqrtint( 40*n + 9); polcoeff( sum( k=(-s + 6)\10, (s - 3)\10, (-1)^k * x^((5*k + 3)*k/2), A) / sum( k=(-s + 6)\10, (s - 3)\10, x^((5*k + 3)*k/2), A), n))}
%Y A215594 Cf. A080054, A098151.
%K A215594 sign,easy
%O A215594 0,2
%A A215594 _Michael Somos_, Aug 16 2012