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A073384 Seventh convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

%I A073384 #14 Oct 03 2022 04:44:58
%S A073384 1,16,152,1104,6756,36624,181224,834768,3628746,15035504,59829704,
%T A073384 229977904,857894388,3117321456,11067753144,38492230704,131417200419,
%U A073384 441252045408,1459330704656,4760342849504
%N A073384 Seventh convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
%H A073384 G. C. Greubel, <a href="/A073384/b073384.txt">Table of n, a(n) for n = 0..1000</a>
%H A073384 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16,-104,336,-476,-112,1064,-432,-1222,432,1064, 112,-476,-336,-104,-16,-1).
%F A073384 a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073383(k).
%F A073384 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+7, 7) * binomial(n-k, k).
%F A073384 a(n) = ((34083315 +46659654*n +24858030*n^2 +6632968*n^3 +939632*n^4 +67304*n^5 + 1912*n^6)*(n+1)*U(n+1) + (7204365 +13225068*n +8230910*n^2 +2411744*n^3 + 362968*n^4 +27088*n^5 +792*n^6)*(n+2)*U(n))/(2^18*3^2*5*7), with U(n) = A000129(n+1), n >= 0.
%F A073384 G.f.: 1/(1-(2+x)*x)^8.
%F A073384 a(n) = F'''''''(n+8, 2)/7!, that is, 1/7! times the 7th derivative of the (n+8)-th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006
%t A073384 CoefficientList[Series[1/(1-2*x-x^2)^8, {x,0,40}], x] (* _G. C. Greubel_, Oct 03 2022 *)
%o A073384 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^8 )); // _G. C. Greubel_, Oct 03 2022
%o A073384 (SageMath)
%o A073384 def A073384_list(prec):
%o A073384     P.<x> = PowerSeriesRing(ZZ, prec)
%o A073384     return P( 1/(1-2*x-x^2)^8 ).list()
%o A073384 A073384_list(40) # _G. C. Greubel_, Oct 03 2022
%Y A073384 Eighth (m=7) column of triangle A054456, A073383.
%Y A073384 Cf. A000129.
%K A073384 nonn,easy
%O A073384 0,2
%A A073384 _Wolfdieter Lang_, Aug 02 2002