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%I #21 Sep 08 2022 08:45:07
%S 1,1,0,-1,0,2,2,-1,-3,0,5,4,-4,-8,1,13,8,-13,-20,6,34,15,-39,-48,25,
%T 88,24,-112,-111,89,224,23,-312,-246,290,559,-43,-848,-515,892,1364,
%U -376,-2255,-987,2632,3243,-1644,-5874,-1598,7519,7473,-5920,-14991,-1552,20912,16544,-19359,-37455,2816,56815
%N Expansion of 1/( (1-x)*(1 + x^2 + x^3) ).
%C The Gi1 sums, see A180662 for the definition of these sums, of triangle A101950 equal the terms of this sequence. - _Johannes W. Meijer_, Aug 06 2011
%H Michael De Vlieger, <a href="/A077889/b077889.txt">Table of n, a(n) for n = 0..10000</a>
%H Eunmi Choi, Yuna Oh, <a href="https://doi.org/10.11568/kjm.2019.27.3.723">Diagonal sums in negative trinomial table</a>, Korean J. Math (2019) Vol. 27, No. 3, 723-734.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,0,1).
%F a(n) = Sum_{k=0..floor(n/4)} A101950(n-3*k, k).
%F a(n) = (1 + 2*A077962(n) + 2*A077962(n-1) + A077962(n-2))/3. - _G. C. Greubel_, Dec 30 2019
%F a(n)-a(n-1) = A077962(n). - _R. J. Mathar_, Mar 14 2021
%p A101950 := proc(n,k) local j,k1: add((-1)^((n-j)/2)*binomial((n+j)/2,j)*(1+(-1)^(n+j))* binomial(j,k)/2, j=0..n) end: A077889 := proc(n): add(A101950(n-3*k,k), k=0..floor(n/4)) end: seq(A077889(n), n=0..60); # _Johannes W. Meijer_, Aug 06 2011
%t CoefficientList[Series[1/((1-x)*(1+x^2+x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{1,-1,0,1},{1,1,0,-1},60] (* _Harvey P. Dale_, Jul 14 2017 *)
%o (PARI) my(x='x+O('x^60)); Vec(1/((1-x)*(1+x^2+x^3))) \\ _G. C. Greubel_, Dec 30 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1+x^2+x^3)) )); // _G. C. Greubel_, Dec 30 2019
%o (Sage)
%o def A077952_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/((1-x)*(1+x^2+x^3)) ).list()
%o A077952_list(60) # _G. C. Greubel_, Dec 30 2019
%o (GAP) a:=[1,1,0,-1];; for n in [5..60] do a[n]:=a[n-1]-a[n-2]+a[n-4]; od; a; # _G. C. Greubel_, Dec 30 2019
%Y Cf. A077962, A101950.
%K sign,easy
%O 0,6
%A _N. J. A. Sloane_, Nov 17 2002
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