This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027991 #19 Aug 08 2022 11:56:29
%S A027991 1,3,12,40,130,404,1227,3653,10720,31090,89316,254568,720757,2029095,
%T A027991 5684340,15855964,44061862,122032508,336966015,927953705,2549229256,
%U A027991 6987648358,19115124552,52194037200,142274514025,387215773899
%N A027991 a(n) = Sum{T(n,k)*T(n,2n-k)}, 0<=k<=n-1, T given by A027926.
%C A027991 From _Wolfdieter Lang_, Jan 02 2012: (Start)
%C A027991 a(n) = A024458(2*n-1), n>=1 (bisection, odd arguments).
%C A027991 chate(n):=a(n+1), n>=0, is the even part of the bisection of the half-convolution of the sequence A000045(n+1), n>=0, with itself. See a comment on A201204 for the definition of half-convolution. There one finds also the rule for the o.g.f.s of the bisection. Here the o.g.f. of the sequence chate(n), n>=0, is Chate(x):= (Ce(x)+U2(x))/2 with Ce(x)=(1-x+x^2)/(1-3*x+x^2)^2, the o.g.f. of A054444(n), and
%C A027991 U2(x)=(1-x)/((1+x)*(1-3*x+x^2)), the o.g.f. of A007598(n+1), n>=0. This results (after multiplying with x) in the o.g.f. given below in the formula section. It is equivalent to the explicit formula given there, as can be seen after a partial fraction decomposition of the o.g.f.
%C A027991 (End)
%F A027991 a(n) = (1/5)[n*F(2n+2) - n*F(2n-2) + F(2n-1) - (-1)^n], F(n)=A000045(n).
%F A027991 O.g.f.: x*(1-2*x+2*x^2)/((1-3*x+x^2)^2*(1+x)). See the comment above. - _Wolfdieter Lang_, Jan 02 2012
%Y A027991 Cf. A027926, A024458, A201204, A007598.
%K A027991 nonn
%O A027991 1,2
%A A027991 _Clark Kimberling_