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%I #16 Apr 08 2021 11:53:21
%S 1,2,9,63,621,8127,135729,2810403,70558101,2109957687,74061977049,
%T 3014272078443,140764140297981,7474792551154047,447790997859123969,
%U 30053688313164013683,2245843775591721612261,185829940905166760571207,16941047558158020804504489
%N a(1) = 1; for n>1, a(n) = (c(n) + c(n+1))/2, where c(n) = A130168(n).
%H 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 G.f.: (1+x)^2/(3*x^3)*Q(0) + (x^3 - 5*x^2 - 5*x - 2)/(6*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
%t b[n_] := (-2^(-1))^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))* BernoulliB[n+k+1], {k, 0, n}];
%t c[n_] := (b[n] + b[n+1])/3;
%t a[n_] := If[n == 1, 1, (c[n] + c[n+1])/2];
%t a /@ Range[1, 19] (* _Jean-François Alcover_, Apr 08 2021 *)
%Y Cf. A000366, A130168.
%K nonn
%O 1,2
%A _Don Knuth_, Aug 02 2007
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