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A073381 Fourth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

%I A073381 #27 May 07 2025 09:13:06
%S A073381 1,10,65,340,1555,6482,25235,93200,330070,1129580,3756950,12197320,
%T A073381 38787770,121148300,372476410,1129367632,3382133695,10016694470,
%U A073381 29370557375,85341915260,245939376949,703423066190
%N A073381 Fourth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
%H A073381 G. C. Greubel, <a href="/A073381/b073381.txt">Table of n, a(n) for n = 0..1000</a>
%H A073381 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H A073381 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,40,30,-68,-30,40,35,10,1).
%F A073381 a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073380(k).
%F A073381 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+4, 4) * binomial(n-k, k).
%F A073381 a(n) = ((2457 +2128*n +572*n^2 +48*n^3)*(n+1)*U(n+1) + 5*(123 +142*n +44*n^2 +4*n^3) *(n+2)*U(n))/(3*2^11), with U(n) = A000129(n+1), n>=0.
%F A073381 G.f.: 1/(1-(2+x)*x)^5.
%F A073381 a(n) = F''''(n+5, 2)/4!, that is, 1/4! times the 4th derivative of the (n+5)th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006
%t A073381 CoefficientList[Series[1/(1-2*x-x^2)^5, {x,0,40}], x] (* _G. C. Greubel_, Oct 02 2022 *)
%o A073381 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^5 )); // _G. C. Greubel_, Oct 02 2022
%o A073381 (SageMath)
%o A073381 def A073381_list(prec):
%o A073381     P.<x> = PowerSeriesRing(ZZ, prec)
%o A073381     return P( 1/(1-2*x-x^2)^5 ).list()
%o A073381 A073381_list(40) # _G. C. Greubel_, Oct 02 2022
%Y A073381 Fifth (m=4) column of triangle A054456, A073380.
%Y A073381 Cf. A000129.
%K A073381 nonn,easy
%O A073381 0,2
%A A073381 _Wolfdieter Lang_, Aug 02 2002