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 A375565 #14 Oct 17 2024 09:57:52
%S A375565 1,2,3,6,17,42,90,194,441,1006,2242,4950,10974,24376,53961,119048,
%T A375565 262337,577782,1271117,2792718,6129342,13441616,29454517,64492800,
%U A375565 141108878,308542280,674238780,1472532300,3214268735,7012637490,15292425923,33333338466,72627184389
%N A375565 a(n) = Sum_{k=0..floor(n/3)} (n-2*k+1) * binomial(n-2*k,k)^2.
%F A375565 G.f.: (1-x-x^3)/((1-x-x^3)^2 - 4*x^4)^(3/2).
%F A375565 D-finite with recurrence 3*n*(n-1)*a(n) -(8*n-3)*(n-1)*a(n-1) +(7*n^2-14*n+8)*a(n-2) +(-8*n^2+3*n+23)*a(n-3) -2*n*(n+8)*a(n-4) +4*((n-1)^2)*a(n-5) +3*n*(n+2)*a(n-6) -2*n*(n-1)*a(n-7)=0. - _R. J. Mathar_, Oct 17 2024
%p A375565 A375565 := proc(n)
%p A375565 add((n-2*k+1)*binomial(n-2*k,k)^2,k=0..floor(n/3)) ;
%p A375565 end proc:
%p A375565 seq(A375565(n),n=0..80) ; # _R. J. Mathar_, Oct 17 2024
%o A375565 (PARI) a(n) = sum(k=0, n\3, (n-2*k+1)*binomial(n-2*k, k)^2);
%Y A375565 Cf. A182884, A376735.
%Y A375565 Cf. A246840.
%K A375565 nonn
%O A375565 0,2
%A A375565 _Seiichi Manyama_, Oct 17 2024