A073385 - OEIS

%I #28 Oct 03 2022 08:45:36

%S 1,18,189,1500,9945,58014,307197,1507176,6950295,30443270,127666539,

%T 515754252,2017069431,7667214570,28419251715,102997948704,

%U 365832349542,1275914693196,4376992440590

%N Eighth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

%C For a(n) in terms of U(n+1) and U(n) with U(n) = A000129(n+1) see the row polynomials of triangles A058402 and A058403 and the comment there.

%H G. C. Greubel, <a href="/A073385/b073385.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (18,-135,528,-1044,504,1764,-2448,-1422,3308,1422, -2448,-1764,504,1044,528,135,18,1).

%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073384(k).

%F a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k)*binomial(n-k+8, 8)*binomial(n-k, k).

%F G.f.: 1/(1-(2+x)*x)^9.

%F a(n) = F''''''''(n+9, 2)/8!, that is, 1/8! times the 8th derivative of the (n+9)-th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006

%t CoefficientList[Series[1/(1-(2+x)x)^9,{x,0,20}],x] (* _Harvey P. Dale_, Apr 26 2017 *)

%o (Sage) taylor( 1/(1-2*x-x^2)^9, x, 0,27).list() # _G. C. Greubel_, Oct 03 2022

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^9 )); // _G. C. Greubel_, Oct 03 2022

%Y Ninth (m=8) column of triangle A054456.

%Y Cf. A000129, A058402, A058403, A073384.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002