A293269 G.f.: Im(1/(1 + i*x/(1 + i*x^2/(1 + i*x^3/(1 + i*x^4/(1 + i*x^5/(1 + ...))))))), a continued fraction, where i is the imaginary unit.
0, -1, 0, 1, 2, 0, -3, -5, -1, 7, 12, 3, -16, -27, -7, 37, 64, 20, -85, -152, -55, 191, 356, 141, -436, -841, -364, 991, 1988, 938, -2233, -4674, -2369, 5044, 11004, 5963, -11361, -25898, -14959, 25467, 60821, 37245, -56995, -142783, -92384, 127136, 334946, 228385, -282392
Offset: 0
Keywords
Examples
G.f. A(x) = Sum_{n>=0} (A293268(n) + i*a(n))*x^n = 1 - i*x - x^2 - (1 - i)*x^3 + (1 + 2*i)*x^4 + 3*x^5 + (2 - 3*i)*x^6 - (2 + 5*i)*x^7 - (7 + i)*x^8 - ...
Links
- Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
Programs
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Mathematica
nmax = 48; Im[CoefficientList[Series[1/(1 + ContinuedFractionK[I x^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x]] nmax = 48; Im[CoefficientList[Series[Sum[I^k x^(k (k + 1)) / Product[1 - x^m, {m, 1, k}], {k, 0, nmax}] / Sum[I^k x^(k^2) / Product[1 - x^m, {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]]
Formula
G.f.: Im( (Sum_{k>=0} i^k*x^(k*(k+1))/Product_{m=1..k} (1 - x^m)) / (Sum_{k>=0} i^k*x^(k^2)/Product_{m=1..k} (1 - x^m)) ), where i is the imaginary unit.