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Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Denominators.
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%I #20 Feb 08 2023 06:05:46

%S 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 3,1,435,1,7161,1,255,1,3,1,959595,1,3,1,6765,1,903,1,345,1,141,1,

%U 23205,1,33,1,795,1,399,1

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

%C Fractions: 1/2, 1/2, -4/3, 2, -38/15, 3, -73/21, 4, -68/15, 5, -179/33, 6, -9218/1365, 7, ... .

%C Numerators are A307974(n).

%C a(2n) same as denominators of cosecant numbers A001897 for n>0 (conjectured).

%F a(n) = A141459(n) * A141044(n).

%F a(n) = A141459(n) for n>2.

%F a(2n+1) = A054977(n).

%F a(2n) = A001897(n) * A054977(n).

%t b[n_] = BernoulliB[n]; b[0] = 1/2; b[1] = 1;

%t a[n_] := Sum[(-1)^(n - k)*Binomial[n, k]*b[k], {k, 0, m}] // Denominator;

%t Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, Jun 04 2019 *)

%Y Essentially the same as A141459.

%Y Cf. A001897, A027642, A054977, A141044, A164555, A307974.

%K nonn,frac

%O 0,1

%A _Paul Curtz_, Jun 04 2019