A291333 a(n) = [x^n] 1/(1 - x/(1 - 2^n*x/(1 - 3^n*x/(1 - 4^n*x/(1 - 5^n*x/(1 - ...)))))), a continued fraction.

1, 1, 5, 297, 485729, 38103228225, 220579355255364545, 134210828762693919568092033, 11583583466188874003924403353591815169, 183988806081826466732185672966967145613350641690625, 676960735217941793634104089611911809588055950029181968418342810625

Offset: 0

Author

Ilya Gutkovskiy, Aug 22 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..30

Crossrefs

Main diagonal of A290569.

Cf. A036740.

Programs

Maple

seq(coeff(series(numtheory:-cfrac([0,[1,1],seq([-i^n*x,1],i=1..n)]),x,n+1),x,n),n=0..15); # Robert Israel, Aug 22 2017

Mathematica

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-i^n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 10}]

Formula

a(n) = A290569(n,n).

a(n) ~ c * (n!)^n ~ c * 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12), where c = 1/QPochhammer(exp(-1)) = 1.9824409074128737036856824655613120156828827... - Vaclav Kotesovec, Aug 26 2017, updated Jul 21 2018