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A130168 a(n) = (b(n) + b(n+1))/3, where b(n) = A000366(n).

%I A130168 #23 Apr 16 2023 13:41:00
%S A130168 1,3,15,111,1131,15123,256335,5364471,135751731,4084163643,
%T A130168 144039790455,5884504366431,275643776229531,14673941326078563,
%U A130168 880908054392169375,59226468571935857991,4432461082611507366531,367227420727722013775883,33514867695588319595233095
%N A130168 a(n) = (b(n) + b(n+1))/3, where b(n) = A000366(n).
%C A130168 As remarked by Gessel, A000366 has a combinatorial interpretation via a certain 2n X n array; this sequence is for a similar array of size (2n-1) X (n-1).
%C A130168 In effect, Dellac gives a combinatorial reason why the elements of A000366 are alternately -1 and +1 modulo 3. Dellac also shows that all the terms of this sequence are odd.
%H A130168 Hippolyte Dellac, <a href="https://odyssee.univ-amu.fr/files/original/2/18/Annales-faculte-sc-Mrs_1901_T-11.pdf">Note sur l'élimination, méthode de parallélogramme</a>, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164. [Warning 76 Mb; go to p. 81 in the pdf file]
%F A130168 G.f.: 2*(1+x)/(3*x^3)*Q(0) - 2/(3*x) - 1/x^2 - 2/(3*x^3), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 2/(1 - x*(k+1)^2/( x*(k+1)^2 - 2/Q(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 22 2013
%F A130168 a(n) = |(2-2^(n+2))*Bernoulli(n+1) - (n+1)*(1-2^(2n+2))*Bernoulli(2n+2) - (1-2^(2n+3))*Bernoulli(2n+3) + Sum_{k=0..n-1} (2*binomial(n,k+1)-binomial(n+1,k))*(1-2^(n+k+2))*Bernoulli(n+k+2)|/(3*2^(n-1)). - _Chai Wah Wu_, Apr 14 2023
%t A130168 b[n_] := (-2^(-1))^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))* BernoulliB[n+k+1], {k, 0, n}];
%t A130168 a[n_] := (b[n] + b[n+1])/3;
%t A130168 a /@ Range[2, 20] (* _Jean-François Alcover_, Apr 08 2021 *)
%o A130168 (Python)
%o A130168 from math import comb
%o A130168 from sympy import bernoulli
%o A130168 def A130168(n): return (abs((2-(2<<n+1))*bernoulli(n+1)-(n+1)*(1-(1<<(m:=n+1<<1)))*bernoulli(m)-(1-(1<<m+1))*bernoulli(m+1)+sum((2*comb(n,k+1)-comb(n+1,k))*(1-(1<<(m:=n+k+2)))*bernoulli(m) for k in range(0,n)))>>n-1)//3 # _Chai Wah Wu_, Apr 14 2023
%Y A130168 Cf. A000366, A130169.
%K A130168 nonn
%O A130168 2,2
%A A130168 _Don Knuth_, Aug 02 2007