A053254 -id:A053254 - cp's OEIS frontend

A291193 Expansion of 1 + x(1-x)/(1 + x^2(1-x^2)/(1 + x^3(1-x^3)/(1 + x^4(1-x^4)/(1 + x^5*(1-x^5)/(1 + ...))))), a continued fraction.

1, 1, -1, -1, 1, 2, -1, -4, 0, 6, 3, -7, -8, 6, 15, -2, -24, -9, 33, 32, -35, -68, 20, 114, 25, -164, -120, 196, 285, -160, -521, -16, 796, 423, -1021, -1166, 999, 2310, -387, -3774, -1296, 5194, 4608, -5735, -10007, 3870, 17441, 2750, -25635, -17116, 31111

Offset: 0

Seiichi Manyama, Aug 20 2017

sign

			G.f. = 1 + x - x^2 - x^3 + x^4 + 2*x^5 - x^6 - 4*x^7 + 6*x^9 + ...

Eric Weisstein's World of Mathematics, Mock Theta Function

Cf. A053254 (nu(q)), A291200.

G.f.: 1/nu(q) where nu(q) is the '3rd order' mock theta function defined by Sum_{n >= 0} q^(n(n+1))/((1+q)(1+q^3)...(1+q^(2n+1))).

a(n) = (-1)^n*A291200(n). - R. J. Mathar, May 16 2024

A308745 Expansion of 1/(1 - x(1 + x)/(1 - x^2(1 + x^2)/(1 - x^3(1 + x^3)/(1 - x^4(1 + x^4)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 36, 76, 161, 342, 726, 1542, 3276, 6960, 14788, 31422, 66767, 141872, 301464, 640584, 1361188, 2892417, 6146164, 13060136, 27751818, 58970564, 125308114, 266270558, 565805452, 1202295228, 2554789536, 5428741218, 11535678790, 24512475453

Offset: 0

Ilya Gutkovskiy, Jun 21 2019

Cf. A005169, A053254, A092848, A143064.

nmax = 33; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^k (1 + x^k), 1, {k, 1, nmax}]), {x, 0, nmax}], x]

From Vaclav Kotesovec, Jun 25 2019: (Start)
a(n) ~ c * d^n, where
d = 2.124927028900893046638236231387101475346473032396641627320401...
c = 0.386397654364351443933577245182777062935616240164642598839093... (End)
From Peter Bala, Dec 18 2020: (Start)
Conjectural g.f.: 1/(2 - (1 + x)/(1 - x^2/(2 - (1 + x^3)/(1 - x^4/(2 - (1 + x^5)/(1 - x^6/(2 - ... ))))))).
More generally it appears that 1/(1 - t*x*(1 + u*x)/(1 - t*x^2*(1 + u*x^2)/(1 - t*x^3*(1 + u*x^3)/(1 - t*x^4*(1 + u*x^4)/(1 - ... ))))) = 1/(1 + u - (u + t*x)/(1 - t*x^2/(1 + u - (u + t*x^3)/(1 - t*x^4/(1 + u - (u + t*x^5)/(1 - ... )))))). (End)

cp's OEIS Frontend

A291193 Expansion of 1 + x(1-x)/(1 + x^2(1-x^2)/(1 + x^3(1-x^3)/(1 + x^4(1-x^4)/(1 + x^5*(1-x^5)/(1 + ...))))), a continued fraction.

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A308745 Expansion of 1/(1 - x(1 + x)/(1 - x^2(1 + x^2)/(1 - x^3(1 + x^3)/(1 - x^4(1 + x^4)/(1 - ...))))), a continued fraction.

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cp's OEIS Frontend

A291193 Expansion of 1 + x*(1-x)/(1 + x^2*(1-x^2)/(1 + x^3*(1-x^3)/(1 + x^4*(1-x^4)/(1 + x^5*(1-x^5)/(1 + ...))))), a continued fraction.

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A308745 Expansion of 1/(1 - x*(1 + x)/(1 - x^2*(1 + x^2)/(1 - x^3*(1 + x^3)/(1 - x^4*(1 + x^4)/(1 - ...))))), a continued fraction.

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A291193 Expansion of 1 + x(1-x)/(1 + x^2(1-x^2)/(1 + x^3(1-x^3)/(1 + x^4(1-x^4)/(1 + x^5*(1-x^5)/(1 + ...))))), a continued fraction.

A308745 Expansion of 1/(1 - x(1 + x)/(1 - x^2(1 + x^2)/(1 - x^3(1 + x^3)/(1 - x^4(1 + x^4)/(1 - ...))))), a continued fraction.