A295703 Expansion of R(x*R(x)), where R(x) = 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction (g.f. for A007325).

1, -1, 2, -3, 2, 4, -18, 43, -80, 123, -148, 78, 287, -1364, 3858, -8627, 15901, -23076, 20061, 18294, -140623, 420241, -930040, 1655753, -2293975, 1872682, 1835066, -12983537, 37871888, -83222132, 149287250, -212064236, 186932259, 131172644, -1139053896, 3449157957, -7710640256

Offset: 0

Author

Ilya Gutkovskiy, Nov 29 2017

Keywords

• sign

Links

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

Crossrefs

Cf. A003823, A007325, A055101, A055102, A055103, A127632, A286509, A291651.

Programs

Mathematica

nmax = 36; CoefficientList[Series[1/(1 + ContinuedFractionK[(x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
g[x_] := g[x] = QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]); a[n_] := a[n] = SeriesCoefficient[g[x g[x]], {x, 0, n}];  Table[a[n], {n, 0, 36}]

Formula

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