A285637 - OEIS

A285637

G.f.: 1/( (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) * (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))) ), a continued fraction.

1, 0, 1, 2, 1, 4, 6, 10, 19, 30, 55, 92, 161, 282, 483, 846, 1462, 2538, 4409, 7642, 13276, 23032, 39977, 69394, 120426, 209036, 362800, 629698, 1092952, 1896968, 3292522, 5714678, 9918752, 17215620, 29880461, 51862438, 90015657, 156236814, 271174435, 470667300, 816919764, 1417897172, 2460991365

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OFFSET

0,4

LINKS

Table of n, a(n) for n=0..42.

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

FORMULA

G.f.: A(x) = R(x)*P(x)/Q(x), where 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)), P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m) and Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).

a(n) ~ c * d^n, where d = 1/A347901 = 1.7356628245303474256582607497196685302546528472903927546099... and c = 0.215558365582078354136603033062960103377669... - Vaclav Kotesovec, Aug 26 2017

EXAMPLE

G.f.: A(x) = 1 + x^2 + 2*x^3 + x^4 + 4*x^5 + 6*x^6 + 10*x^7 + 19*x^8 + 30*x^9 + 55*x^10 + ...

MATHEMATICA

nmax = 42; 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 = 42; CoefficientList[Series[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 (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A003823, A005169, A007325, A055101, A285635, A285636, A285638.

Sequence in context: A283309 A366043 A054408 * A205845 A034424 A095012

Adjacent sequences: A285634 A285635 A285636 * A285638 A285639 A285640

KEYWORD

nonn,changed

AUTHOR

Ilya Gutkovskiy, Apr 23 2017

STATUS

approved