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A102590 Inverse Boustrophedon transform of 2^n.

%I A102590 #19 Jun 11 2022 03:32:50
%S A102590 1,1,1,0,-3,-14,-39,-130,-263,-1214,-179,-21810,98277,-1021214,
%T A102590 8446881,-82814290,836117617,-9075846014,103898533141,-1257148371570,
%U A102590 16004750729757,-213975589371614,2996827456610601,-43880489398997650,670443584312526697,-10670445866332254014
%N A102590 Inverse Boustrophedon transform of 2^n.
%C A102590 Binomial transform of (-1)^n*A062162.
%H A102590 Alois P. Heinz, <a href="/A102590/b102590.txt">Table of n, a(n) for n = 0..200</a>
%F A102590 E.g.f.: exp(2x)/(sec(x)+tan(x)) = cos(x)exp(2x)/(1+sin(x)).
%F A102590 a(n) ~ (-1)^n * n^(n+1/2)*2^(n+5/2)/(Pi^(n+1/2)*exp(n+Pi)). - _Vaclav Kotesovec_, Sep 29 2013
%F A102590 G.f.: E(0)*x/(x-1)/(1-2*x) + 1/(1-2*x), where E(k) = 1 - x^2*(k+1)*(k+2)/( x^2*(k+1)*(k+2) - 2*(x*(k-1)+1)*(x*k+1)/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2014
%p A102590 a:= n-> n!*coeff(series(exp(2*x)/(sec(x)+tan(x)), x, n+1), x, n):
%p A102590 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 29 2013
%t A102590 CoefficientList[Series[Cos[x]*E^(2*x)/(1+Sin[x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 29 2013 *)
%o A102590 (Python)
%o A102590 from itertools import islice, accumulate
%o A102590 from operator import sub
%o A102590 def A102590_gen(): # generator of terms
%o A102590     blist, m = tuple(), 1
%o A102590     while True:
%o A102590         yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=m)))[-1]
%o A102590         m *= 2
%o A102590 A102590_list = list(islice(A102590_gen(),20)) # _Chai Wah Wu_, Jun 10 2022
%Y A102590 Cf. A000079, A062162.
%K A102590 easy,sign
%O A102590 0,5
%A A102590 _Paul Barry_, Jan 22 2005