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A144413 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4).

%I A144413 #20 Mar 27 2021 22:41:10
%S A144413 0,1,-1,1,0,-3,10,-24,49,-89,145,-208,245,-174,-176,1121,-3185,7137,
%T A144413 -13920,24301,-37926,51256,-53615,20407,97265,-386224,984549,-2083934,
%U A144413 3896480,-6537023,9734175,-12231999,10690624,2126301,-39992150,126414472,-297132815,598577351,-1075051951,1730868336,-2443923755
%N A144413 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4).
%H A144413 G. C. Greubel, <a href="/A144413/b144413.txt">Table of n, a(n) for n = 0..500</a>
%H A144413 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2,1).
%F A144413 From _R. J. Mathar_, Jan 21 2009: (Start)
%F A144413 a(n) = -3*a(n-1) - 2*a(n-2) + a(n-3).
%F A144413 G.f.: x*(1 +2*x)/(1 +3*x +2*x^2 -x^3). (End)
%t A144413 (* First program *)
%t A144413 b[n_]:= b[n]= If[n==0, 0, If[n<3, 1, b[n-2] +b[n-3]]];
%t A144413 a[n_]:= Sum[(-1)^m*Binomial[n, m]*b[n-m], {m,0,n}];
%t A144413 Table[a[n], {n, 0, 50}]
%t A144413 (* Second program *)
%t A144413 LinearRecurrence[{-3,-2,1}, {0,1,-1}, 51] (* _G. C. Greubel_, Mar 27 2021 *)
%o A144413 (Magma) I:=[0,1,1]; [n le 3 select I[n] else -3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..51]]; // _G. C. Greubel_, Mar 27 2021
%o A144413 (Sage)
%o A144413 @CachedFunction
%o A144413 def A000931(n): return 1 if n==0 else sum( binomial(k, n-2*k-3) for k in (0..floor((n-3)/2)))
%o A144413 def A144413(n): return sum( (-1)^k*binomial(n, k)*A000931(n-k+4) for k in (0..n))
%o A144413 [A144413(n) for n in (0..50)] # _G. C. Greubel_, Mar 27 2021
%Y A144413 Cf. A000073, A000931, A073358.
%K A144413 sign,easy
%O A144413 0,6
%A A144413 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 30 2008
%E A144413 Terms a(30) onward added and edited by _G. C. Greubel_, Mar 27 2021