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%I #36 Sep 08 2022 08:45:09
%S 0,1,8,50,288,1604,8800,47944,260352,1411600,7647872,41420576,
%T 224294400,1214467136,6575615488,35602384000,192760455168,
%U 1043650265344,5650555750400,30593342288384,165638957801472,896804870374400
%N 4th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).
%H Vincenzo Librandi, <a href="/A081180/b081180.txt">Table of n, a(n) for n = 0..300</a>
%H S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-14).
%F a(n) = 8a(n-1) - 14a(n-2), a(0)=0, a(1)=1.
%F G.f.: x/(1 - 8x + 14x^2).
%F a(n) = ((4 + sqrt(2))^n - (4 - sqrt(2))^n/(2*sqrt(2)).
%F a(n) = Sum_{k=0..n} C(n,2k+1) 2^k*4^(n-2k-1).
%F If shifted once left, fourth binomial transform of A143095. - Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009, _R. J. Mathar_, Oct 15 2009
%F E.g.f.: exp(4*x)*sinh(sqrt(2)*x)/sqrt(2). - _Ilya Gutkovskiy_, Aug 12 2017
%t Join[{a=0,b=1},Table[c=8*b-14*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 19 2011 *)
%t CoefficientList[Series[x / (1 - 8 x + 14 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%t LinearRecurrence[{8,-14},{0,1},30] (* _Harvey P. Dale_, Aug 17 2019 *)
%o (Sage) [lucas_number1(n,8,14) for n in range(0, 22)] # _Zerinvary Lajos_, Apr 23 2009
%o (Magma) I:=[0, 1]; [n le 2 select I[n] else 8*Self(n-1)-14*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 06 2013
%Y Binomial transform of A081179.
%Y Cf. A081182.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Mar 11 2003
%E Modified the completing comment on the fourth binomial transform - _R. J. Mathar_, Oct 15 2009
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