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A073389 Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

%I A073389 #27 Oct 04 2022 08:38:12
%S A073389 1,6,30,128,504,1872,6672,23040,77616,256288,832416,2666496,8441600,
%T A073389 26454528,82174464,253280256,775316736,2358812160,7137023488,
%U A073389 21487386624,64401106944,192229535744,571630694400,1693996941312,5004131659776,14738997288960,43293528760320
%N A073389 Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
%H A073389 Muniru A Asiru, <a href="/A073389/b073389.txt">Table of n, a(n) for n = 0..500</a>
%H A073389 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6,-16,12,24,8).
%F A073389 a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073388(k).
%F A073389 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2)*binomial(n-k, k)*2^(n-k).
%F A073389 a(n) = (n+3)*((n+1)*U(n+1) + (n+2)*U(n))/12, with U(n) = A002605(n), n >= 0.
%F A073389 G.f.: 1/(1-2*x*(1+x))^3.
%t A073389 CoefficientList[Series[1/(1-2x(1+x))^3,{x,0,25}],x]  (* _Harvey P. Dale_, Mar 14 2011 *)
%o A073389 (Sage) taylor( 1/(1-2*x-2*x^2)^3, x, 0, 25).list() # _Zerinvary Lajos_, Jun 03 2009; modified by _G. C. Greubel_, Oct 03 2022
%o A073389 (GAP) List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+2,2)*Binomial(n-k,k)*(1/2)^k)); # _Muniru A Asiru_, Jun 12 2018
%o A073389 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^3 )); // _G. C. Greubel_, Oct 03 2022
%Y A073389 Third (m=2) column of triangle A073387, A073388.
%Y A073389 Cf. A002605.
%K A073389 nonn,easy
%O A073389 0,2
%A A073389 _Wolfdieter Lang_, Aug 02 2002