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A027010 a(n) = Sum_{k=floor((n+1)/2)..n} T(k,n-k); i.e., a(n) is the n-th diagonal sum of left-justified array T given by A026998.

%I A027010 #16 Jul 15 2025 17:12:51
%S A027010 1,1,2,5,6,13,17,29,43,64,100,144,223,326,492,733,1089,1634,2421,3626,
%T A027010 5389,8041,11985,17847,26624,39640,59112,88059,131242,195592,291433,
%U A027010 434369,647218,964581,1437374,2142013,3192113,4756821
%N A027010 a(n) = Sum_{k=floor((n+1)/2)..n} T(k,n-k); i.e., a(n) is the n-th diagonal sum of left-justified array T given by A026998.
%H A027010 Vincenzo Librandi, <a href="/A027010/b027010.txt">Table of n, a(n) for n = 1..1000</a>
%H A027010 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-2,1).
%F A027010 G.f.: x*(1 - x^2 + 2*x^3)/((1-x)*(1-2*x^2-x^3+x^4)).
%F A027010 a(n) = 2*b(n+2) + 3*b(n+1) - b(n) - 4*b(n-1) - 2, where b(n) = A122514(n). - _G. C. Greubel_, Jul 11 2025
%t A027010 CoefficientList[Series[(1-x^2+2 x^3)/((1-x)(1-2 x^2 -x^3 +x^4)), {x,0,40}], x] (* _Vincenzo Librandi_, Aug 03 2017 *)
%o A027010 (Magma)
%o A027010 R<x>:= PowerSeriesRing(Integers(), 40);
%o A027010 Coefficients(R!( x*(1-x^2+2*x^3)/((1-x)*(1-2*x^2-x^3+x^4)) )); // _G. C. Greubel_, Jul 11 2025
%o A027010 (SageMath)
%o A027010 def A027010_list(prec):
%o A027010     P.<x> = PowerSeriesRing(ZZ, prec)
%o A027010     return P( x*(1-x^2+2*x^3)/((1-x)*(1-2*x^2-x^3+x^4)) ).list()
%o A027010 a=A027010_list(40); a[1:] # _G. C. Greubel_, Jul 11 2025
%Y A027010 Cf. A026998, A122514.
%K A027010 nonn
%O A027010 1,3
%A A027010 _Clark Kimberling_