%I #33 Sep 08 2022 08:45:09

%S 0,1,16,194,2112,21764,217280,2127112,20562432,197117968,1879016704,

%T 17842953248,168988216320,1597548359744,15083504344064,

%U 142288071200896,1341431869882368,12641049503662336,119088016125890560

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

%C This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - _G. C. Greubel_, Mar 12 2021

%H Vincenzo Librandi, <a href="/A081185/b081185.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-62).

%F a(n) = 16*a(n-1) - 62*a(n-2), a(0)=0, a(1)=1.

%F G.f.: x/(1 - 16*x + 62*x^2).

%F a(n) = ((8 + sqrt(2))^n - (8 - 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(8*x)*sinh(sqrt(2)*x)/sqrt(2). - _Ilya Gutkovskiy_, Aug 12 2017

%F From _G. C. Greubel_, Mar 12 2021: (Start)

%F a(n) = 2*A147959(n) + 8*A081185(n).

%F a(n) = (1/2)*Sum_{k=0..n-1} binomial(n-1,k)*7^(n-k-1)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

%p m:=30; S:=series( x/(1-16*x+62*x^2), x, m+1):

%p seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 12 2021

%t Join[{a=0,b=1},Table[c=16*b-62*a;a=b;b=c,{n,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2011 *)

%t CoefficientList[Series[x/(1-16x+62x^2), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)

%t LinearRecurrence[{16,-62},{0,1},30] (* _Harvey P. Dale_, Sep 24 2013 *)

%o (Magma) [n le 2 select n-1 else 16*Self(n-1)-62*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Aug 07 2013

%o (Sage) [( x/(1-16*x+62*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Mar 12 2021

%Y Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), this sequence (m=7), A164600 (m=8).

%Y Binomial transform of A081184.

%Y Cf. A081185, A147959.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Mar 11 2003