cp's OEIS Frontend

A073386 Ninth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

%I A073386 #17 Nov 20 2022 11:37:34
%S A073386 1,20,230,1980,14135,88264,497860,2591160,12630475,58295380,256887774,
%T A073386 1087825180,4449607565,17654254880,68177369040,257006941664,
%U A073386 948023601910,3428968838680,12182953719860
%N A073386 Ninth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
%C A073386 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 A073386 G. C. Greubel, <a href="/A073386/b073386.txt">Table of n, a(n) for n = 0..1000</a>
%H A073386 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (20,-170,780,-1965,2064,1800,-6480,1710,8600,-3772, -8600,1710,6480,1800,-2064,-1965,-780,-170,-20,-1).
%F A073386 a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073385(k).
%F A073386 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k)*binomial(n-k+9, 9)*binomial(n-k, k).
%F A073386 G.f.: 1/(1-(2+x)*x)^10.
%F A073386 a(n) = F'''''''''(n+10, 2)/9!, that is, 1/9! times the 9th derivative of the (n+10)th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006
%t A073386 CoefficientList[Series[1/(1-2*x-x^2)^10, {x,0,40}], x] (* _G. C. Greubel_, Oct 03 2022 *)
%t A073386 LinearRecurrence[{20,-170,780,-1965,2064,1800,-6480,1710,8600,-3772,-8600,1710,6480,1800,-2064,-1965,-780,-170,-20,-1},{1,20,230,1980,14135,88264,497860,2591160,12630475,58295380,256887774,1087825180,4449607565,17654254880,68177369040,257006941664,948023601910,3428968838680,12182953719860,42585118702280},20] (* _Harvey P. Dale_, Nov 20 2022 *)
%o A073386 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^10 )); // _G. C. Greubel_, Oct 03 2022
%o A073386 (SageMath)
%o A073386 def A073386_list(prec):
%o A073386     P.<x> = PowerSeriesRing(ZZ, prec)
%o A073386     return P( 1/(1-2*x-x^2)^10 ).list()
%o A073386 A073386_list(40) # _G. C. Greubel_, Oct 03 2022
%Y A073386 Tenth (m=9) column of triangle A054456.
%Y A073386 Cf. A000129, A058402, A058403, A073385.
%K A073386 nonn,easy
%O A073386 0,2
%A A073386 _Wolfdieter Lang_, Aug 02 2002