%I #38 Jan 14 2024 02:29:07

%S 0,1,14,149,1428,12989,114730,995737,8548008,72872473,618458246,

%T 5233409213,44200191420,372832446869,3142245259426,26468308629121,

%U 222870793614672,1876180605036721,15791601170624510,132901927952017253

%N 7th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

%H Vincenzo Librandi, <a href="/A081184/b081184.txt">Table of n, a(n) for n = 0..200</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 (14,-47).

%F a(n) = 14*a(n-1) - 47*a(n-2), a(0)=0, a(1)=1.

%F G.f.: x/(1 - 14*x + 47*x^2). [Corrected by _Georg Fischer_, May 15 2019]

%F a(n) = ((7 + sqrt(2))^n - (7 - sqrt(2))^n)/(2*sqrt(2)).

%F a(n) = Sum_{k=0..n} C(n,2*k+1) * 2^k * 7^(n-2*k-1).

%F E.g.f.: exp(7*x)*sinh(sqrt(2)*x)/sqrt(2). - _Ilya Gutkovskiy_, Aug 12 2017

%t CoefficientList[Series[x/(1-14*x+47*x^2), {x,0,30}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)

%t LinearRecurrence[{14,-47},{0,1},30] (* _Harvey P. Dale_, Nov 12 2013 *)

%o (Magma) [n le 2 select n-1 else 14*Self(n-1)-47*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Aug 07 2013

%o (SageMath)

%o A081184=BinaryRecurrenceSequence(14,-47,0,1)

%o [A081184(n) for n in range(31)] # _G. C. Greubel_, Jan 14 2024

%Y Binomial transform of A081183.

%Y Cf. A081185.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Mar 11 2003