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%I #28 Oct 03 2022 08:45:42
%S 1,12,90,532,2709,12432,52808,211248,805374,2951576,10465476,36079848,
%T 121412942,400054032,1293997416,4117416496,12910964103,39956039172,
%U 122193599822,369685154076,1107503284923,3288114790112
%N Fifth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
%H G. C. Greubel, <a href="/A073382/b073382.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-54,100,-15,-168,76,168,-15,-100,-54,-12,-1).
%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073381(k).
%F a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+5, 5) * binomial(n-k, k).
%F a(n) = ((50085 +53006*n +19594*n^2 +3016*n^3 +164*n^4)*(n+1)*U(n+1) +(11355 +16336*n +7042*n^2 +1184*n^3 +68*n^4)*(n+2)*U(n))/(3*5*2^13), with U(n) = A000129(n+1), n >= 0.
%F G.f.: 1/(1-(2+x)*x)^6.
%F a(n) = F'''''(n+6, 2)/5!, that is, 1/5! times the 5th derivative of the (n+6)th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006
%e x^3 + 12*x^4 + 90*x^5 + 532*x^6 + 2709*x^7 + 12432*x^8 + 52808*x^9 + 211248*x^10 + 805374*x^11 +......+ 122193599822*x^21 + 369685154076*x^22 + 1107503284923*x^23 + 3288114790112*x^24 + etc. [_Zerinvary Lajos_, Jun 01 2009]
%t CoefficientList[Series[1/(1-2*x-x^2)^6, {x,0,40}], x] (* _G. C. Greubel_, Oct 02 2022 *)
%o (Sage) taylor( 1/(1-2*x-x^2)^6, x, 0, 24).list() # _G. C. Greubel_, Oct 02 2022
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^6 )); // _G. C. Greubel_, Oct 02 2022
%Y Sixth (m=5) column of triangle A054456, A073381.
%Y Cf. A000129.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 02 2002
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