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%I #17 Jan 23 2024 11:59:11
%S 1,1,3,17,155,2025,34819,743329,18937707,560071193,18844479635,
%T 710440531665,29654234779771,1357326276747721,67589738142784803,
%U 3637403230889380097,210358430818676801675,13009719599952748481145
%N Expansion of 1/(1-x/(1-2x/(1-4x/(1-6x/(1-8x/(1-.... (continued fraction).
%C Hankel transform is A168442.
%F G.f.: 1/(1-x-2x^2/(1-6x-24x^2/(1-14x-80x^2/(1-22x-168x^2/(1-30x-288x^2/(1-... (continued fraction).
%F a(n) = Sum_{k=0..n} A111106(n,k)*2^(n-k). - _Philippe Deléham_, Nov 28 2009
%F a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
%F 1, 1, 0, 0, 0, 0, ...
%F 2, 2, 2, 0, 0, 0, ...
%F 4, 4, 4, 4, 0, 0, ...
%F 6, 6, 6, 6, 6, 0, ...
%F 8, 8, 8, 8, 8, 8, ...
%F ...
%F (where the series (1,2,4,6,8,...) = A004277, positive even integers prefaced with a 1). - _Gary W. Adamson_, Jul 19 2011
%F G.f. A(x) = 1 + x/(G(0)-x) where G(k) = 1 - x*(2*k+2)/G(k+1)); (continued fraction, 1-step).- _Sergei N. Gladkovskii_, Oct 28 2012
%F a(n) ~ 2^(2*n - 3/2) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Jan 23 2024
%t nmax = 20; CoefficientList[1 + x*Series[1/(1 - x + ContinuedFractionK[-2*k*x, 1, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 23 2024 *)
%Y Cf. A004277, A111106, A168442.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Nov 25 2009
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