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A027959 a(n) = Sum_{k=m..n} T(k,n-k), where m = floor((n+1)/2); a(n) is the n-th diagonal-sum of left justified array T given by A027948.

%I A027959 #18 Sep 08 2022 08:44:49
%S A027959 1,1,2,3,5,7,12,16,27,37,59,85,129,192,285,428,634,949,1412,2104,3140,
%T A027959 4671,6973,10378,15478,23058,34362,51216,76305,113736,169465,252561,
%U A027959 376362,560851,835821,1245503,1856132,2765976,4121947
%N A027959 a(n) = Sum_{k=m..n} T(k,n-k), where m = floor((n+1)/2); a(n) is the n-th diagonal-sum of left justified array T given by A027948.
%H A027959 G. C. Greubel, <a href="/A027959/b027959.txt">Table of n, a(n) for n = 1..1000</a>
%H A027959 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1,-3,-1,1).
%F A027959 G.f.: x*(1+x-x^2-x^3+x^4)/((1-x)*(1+x)*(1-2*x^2-x^3+x^4)). - _Colin Barker_, Nov 25 2014
%p A027959 seq(coeff(series(x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)), x, n+1), x, n), n = 1..40); # _G. C. Greubel_, Sep 30 2019
%t A027959 T[n_, k_]:= If[k==n, 1, Sum[Binomial[k+j, 2*j-1], {j, 0, n-k}]]; Table[Sum[T[k, n-k], {k, Floor[(n-1)/2], n}], {n,0,40}] (* _G. C. Greubel_, Sep 30 2019 *)
%o A027959 (PARI) my(x='x+O('x^40)); Vec(x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4))) \\ _G. C. Greubel_, Sep 30 2019
%o A027959 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)) )); // _G. C. Greubel_, Sep 30 2019
%o A027959 (Sage)
%o A027959 def A027959_list(prec):
%o A027959     P.<x> = PowerSeriesRing(ZZ, prec)
%o A027959     return P( x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)) ).list()
%o A027959 a=A027959_list(40); a[1:] # _G. C. Greubel_, Sep 30 2019
%o A027959 (GAP) a:=[1,1,2,3,5,7];; for n in [7..40] do a[n]:=3*a[n-2]+a[n-3] -3*a[n-4]-a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Sep 30 2019
%Y A027959 Cf. A027948.
%K A027959 nonn
%O A027959 1,3
%A A027959 _Clark Kimberling_