A073380 - OEIS

%I #22 Feb 18 2023 18:01:05

%S 1,8,44,200,810,3032,10716,36248,118435,376240,1167720,3553840,

%T 10636180,31375440,91392040,263266512,750922021,2123059448,5955034740,

%U 16584106040,45884989054,126202397032

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

%H G. C. Greubel, <a href="/A073380/b073380.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-20,8,26,-8,-20,-8,-1).

%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A054457(k).

%F a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+3, 3) * binomial(n-k, k).

%F a(n) = ((147 +94*n +14*n^2)*(n+1)*U(n+1) + 3*(15 +12*n +2*n^2)*(n+2)*U(n))/ (3*2^7), with U(n) = A000129(n+1), n >= 0.

%F G.f.: 1/(1-(2+x)*x)^4.

%F a(n) = F'''(n+4, 2)/6, that is, 1/6 times the 3rd derivative of the (n+4)th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006

%t CoefficientList[Series[1/(1-2*x-x^2)^4, {x,0,40}], x] (* _G. C. Greubel_, Oct 02 2022 *)

%t LinearRecurrence[{8,-20,8,26,-8,-20,-8,-1},{1,8,44,200,810,3032,10716,36248},30] (* _Harvey P. Dale_, Feb 18 2023 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^4 )); // _G. C. Greubel_, Oct 02 2022

%o (SageMath)

%o def A073380_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( 1/(1-2*x-x^2)^4 ).list()

%o A073380_list(30) # _G. C. Greubel_, Oct 02 2022

%Y Fourth (m=3) column of triangle A054456, A054457 (m=2).

%Y Cf. A000129.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002