A302015 Expansion of 1/(1 - x/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))), a continued fraction.

1, 1, 0, 0, 1, 0, -1, 0, 1, 0, -1, 1, 1, -2, -1, 2, 0, -2, 2, 3, -3, -3, 4, 0, -7, 3, 9, -5, -7, 10, 4, -17, -1, 21, -7, -21, 21, 19, -36, -13, 47, -5, -56, 36, 64, -69, -54, 104, 15, -147, 41, 177, -115, -168, 221, 116, -344, -15, 442, -159, -481, 422, 443, -736, -280, 1034, -90, -1276, 681

Offset: 0

Author

Ilya Gutkovskiy, Mar 30 2018

Keywords

• sign

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Crossrefs

Antidiagonal sums of A286509.

Cf. A007325, A167750, A302016.

Programs

Mathematica

nmax = 68; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[1/(1 - x QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5])), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4))/((1 - x^(5*k-2))*(1 - x^(5*k-3)))).

a(0) = 1; a(n) = Sum_{k=1..n} A007325(k-1)*a(n-k).