A291678 -id:A291678 - cp's OEIS frontend

A291679 Main diagonal of A291678.

1, 1, 1, -2, -11, -24, -8, 141, 573, 1087, -174, -8700, -31328, -52740, 36387, 534198, 1742445, 2540583, -3626189, -33115232, -97968686, -118497822, 301668764, 2060526393, 5526622320, 5165256226, -23033840842, -127995025736, -310560935969, -193716799472

Offset: 0

Seiichi Manyama, Aug 29 2017

sign

Cf. A291651, A291678.

a(n) = [x^n] (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))^n, a continued fraction.

A302016 Expansion of 1/(1 - x - x^2/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 46, 68, 102, 153, 229, 342, 510, 761, 1136, 1697, 2535, 3786, 5653, 8441, 12605, 18824, 28112, 41981, 62691, 93617, 139800, 208768, 311761, 465564, 695242, 1038226, 1550415, 2315284, 3457489, 5163181, 7710344, 11514102, 17194374, 25676907, 38344147

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

nonn

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

Antidiagonal sums of A291678.

Cf. A003823, A302015.

nmax = 44; CoefficientList[Series[1/(1 - x - x^2/(1 + ContinuedFractionK[x^k, 1, {k, 2, nmax}])), {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[1/(1 - x QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]/(QPochhammer[x, x^5] QPochhammer[x^4, x^5])), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(5*k-2))*(1 - x^(5*k-3))/((1 - x^(5*k-1))*(1 - x^(5*k-4)))).
a(0) = 1; a(n) = Sum_{k=1..n} A003823(k-1)*a(n-k).
a(n) ~ c / r^n, where r = 0.669643458685499460127124120930664114507093547265881... is the root of the equation x*QPochhammer[x^2, x^5]*QPochhammer[x^3, x^5] = QPochhammer[x, x^5]*QPochhammer[x^4, x^5] and c = 0.833333547701931811823757549354805979633827853516233646128015838266... - Vaclav Kotesovec, Jun 08 2019

cp's OEIS Frontend

A291679 Main diagonal of A291678.

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A302016 Expansion of 1/(1 - x - x^2/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))), a continued fraction.

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