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A073390 Third convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

%I A073390 #27 Oct 04 2022 08:38:08
%S A073390 1,8,48,240,1080,4512,17856,67776,248880,889600,3109376,10664448,
%T A073390 35989248,119761920,393676800,1280157696,4122985728,13165099008,
%U A073390 41713192960,131243970560,410315433984,1275348344832
%N A073390 Third convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
%H A073390 Muniru A Asiru, <a href="/A073390/b073390.txt">Table of n, a(n) for n = 0..500</a>
%H A073390 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,-16,56,32,-64,-64,-16).
%F A073390 a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073389(k).
%F A073390 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, 3)*binomial(n-k, k)*2^(n-k).
%F A073390 a(n) = ((64 + 37*n + 5*n^2)*(n+1)*U(n+1) + 4*(11 + 7*n + n^2)*(n+2)*U(n))/(6^3), with U(n) = A002605(n), n >= 0.
%F A073390 G.f.: 1/(1-2*x*(1+x))^4.
%t A073390 CoefficientList[Series[1/(1-2*x-2*x^2)^4, {x,0,40}], x] (* _G. C. Greubel_, Oct 03 2022 *)
%o A073390 (GAP) List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+3,3)*Binomial(n-k,k)*(1/2)^k)); # _Muniru A Asiru_, Jun 12 2018
%o A073390 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^4 )); // _G. C. Greubel_, Oct 03 2022
%o A073390 (SageMath)
%o A073390 def A073390_list(prec):
%o A073390     P.<x> = PowerSeriesRing(ZZ, prec)
%o A073390     return P( 1/(1-2*x-2*x^2)^4 ).list()
%o A073390 A073390_list(40) # _G. C. Greubel_, Oct 03 2022
%Y A073390 Fourth (m=3) column of triangle A073387.
%Y A073390 Cf. A002605, A073389.
%K A073390 nonn,easy
%O A073390 0,2
%A A073390 _Wolfdieter Lang_, Aug 02 2002