A130169 a(1) = 1; for n>1, a(n) = (c(n) + c(n+1))/2, where c(n) = A130168(n).

1, 2, 9, 63, 621, 8127, 135729, 2810403, 70558101, 2109957687, 74061977049, 3014272078443, 140764140297981, 7474792551154047, 447790997859123969, 30053688313164013683, 2245843775591721612261, 185829940905166760571207, 16941047558158020804504489

Offset: 1

Author

Don Knuth, Aug 02 2007

Keywords

• nonn

Links

• Hippolyte Dellac, Note sur l'élimination, méthode de parallélogramme, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164. [Warning 76 Mb; go to p. 81 in the pdf file]

Crossrefs

Cf. A000366, A130168.

Programs

Mathematica

b[n_] := (-2^(-1))^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))* BernoulliB[n+k+1], {k, 0, n}];
c[n_] := (b[n] + b[n+1])/3;
a[n_] := If[n == 1, 1, (c[n] + c[n+1])/2];
a /@ Range[1, 19] (* Jean-François Alcover, Apr 08 2021 *)

Formula

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