OFFSET
0,2
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.
LINKS
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
FORMULA
G.f.: A(x) = P(x)/(R(x)*Q(x)), where P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m), R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3)) and Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).
a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.452642356466453742995961374156022446123012... - Vaclav Kotesovec, Aug 26 2017, updated Sep 24 2020
EXAMPLE
G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 14*x^6 + 22*x^7 + 36*x^8 + 64*x^9 + ...
MATHEMATICA
nmax = 40; CoefficientList[Series[(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}]))/(1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / (Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}] Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}]), {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 23 2017
STATUS
approved