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A306821 Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Denominators.

%I A306821 #20 Feb 08 2023 06:05:46
%S A306821 2,2,3,1,15,1,21,1,15,1,33,1,1365,1,3,1,255,1,399,1,165,1,69,1,1365,1,
%T A306821 3,1,435,1,7161,1,255,1,3,1,959595,1,3,1,6765,1,903,1,345,1,141,1,
%U A306821 23205,1,33,1,795,1,399,1
%N A306821 Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Denominators.
%C A306821 Fractions: 1/2, 1/2, -4/3, 2, -38/15, 3, -73/21, 4, -68/15, 5, -179/33, 6, -9218/1365, 7, ... .
%C A306821 Numerators are A307974(n).
%C A306821 a(2n) same as denominators of cosecant numbers A001897 for n>0 (conjectured).
%F A306821 a(n) = A141459(n) * A141044(n).
%F A306821 a(n) = A141459(n) for n>2.
%F A306821 a(2n+1) = A054977(n).
%F A306821 a(2n) = A001897(n) * A054977(n).
%t A306821 b[n_] = BernoulliB[n]; b[0] = 1/2; b[1] = 1;
%t A306821 a[n_] := Sum[(-1)^(n - k)*Binomial[n, k]*b[k], {k, 0, m}] // Denominator;
%t A306821 Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, Jun 04 2019 *)
%Y A306821 Essentially the same as A141459.
%Y A306821 Cf. A001897, A027642, A054977, A141044, A164555, A307974.
%K A306821 nonn,frac
%O A306821 0,1
%A A306821 _Paul Curtz_, Jun 04 2019