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%I #24 Sep 08 2022 08:45:00
%S 1,1,3,7,18,46,118,303,778,1998,5131,13177,33840,86905,223182,573157,
%T 1471933,3780093,9707713,24930522,64024444,164422126,422254905,
%U 1084399096,2784861432,7151844025,18366756913,47167941348,121132691065
%N Expansion of ( 1-x-x^2 ) / ( 1-2*x-2*x^2+x^3+x^4 ).
%C Diagonal sums of the Riordan matrix ((1-x-x^2)/(1-2*x-x^2), (x-x^2-x^3) / (1-2*x-x^2)) (A190215). - _Emanuele Munarini_, May 10 2011
%H G. C. Greubel, <a href="/A052960/b052960.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1031">Encyclopedia of Combinatorial Structures 1031</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1,-1).
%F G.f.: (1-x-x^2)/(1-2*x-2*x^2+x^3+x^4).
%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
%F a(n) = Sum_{alpha=RootOf(1-2*z-2*z^2+z^3+z^4)} (1/331)*(25 + 75*alpha - 6*alpha^2 - 5*alpha^3)*alpha^(-1-n).
%F a(n) = Sum_{i=0..n} Sum_{k=0..n/2} binomial(i+2*k, 2*k)*binomial(i+k, n-i-2*k). - _Emanuele Munarini_, May 10 2011
%p spec:= [S,{S=Sequence(Prod(Union(Sequence(Union(Prod(Z,Z),Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p seq(coeff(series((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 23 2019
%t Table[Sum[Sum[Binomial[i+2k,2k]Binomial[i+k,n-i-2k],{k,0,n/2}],{i,0,n}],{n,0,12}] (* _Emanuele Munarini_, May 10 2011 *)
%t LinearRecurrence[{2,2,-1,-1}, {1,1,3,7}, 30] (* _G. C. Greubel_, Oct 23 2019 *)
%t CoefficientList[Series[(1-x-x^2)/(1-2x-2x^2+x^3+x^4),{x,0,50}],x] (* _Harvey P. Dale_, Jan 21 2021 *)
%o (Maxima) makelist(sum(sum(binomial(i+2*k, 2*k)*binomial(i+k, n-i-2*k), k,0,n/2),i,0,n),n,0,24); # _Emanuele Munarini_, May 10 2011
%o (PARI) my(x='x+O('x^30)); Vec((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4)) \\ _G. C. Greubel_, Oct 23 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x-x^2)/(1-2*x-2*x^2+x^3+x^4) )); // _G. C. Greubel_, Oct 23 2019
%o (Sage)
%o def A052960_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4)).list()
%o A052960_list(30) # _G. C. Greubel_, Oct 23 2019
%o (GAP) a:=[1,1,3,7];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3] -a[n-4]; od; a; # _G. C. Greubel_, Oct 23 2019
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Feb 06 2000
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