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COMMENTS
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a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.
The numbers are
BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..
Difference table:
1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42,...
-1/2, -2/3, -1/3, 11/15, 19/15, -47/21, -163/21,...
-1/6, 1/3, 16/15, 8/15, -368/105, -116/21, 2152/105,...
1/2, 11/15, -8/15, -424/105, -212/105, 2732/105, 4204/105,...
7/30, -19/15, -368/195, 212/105, 2944/105, 1472/105,...
-3/2, -47/21, 116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .
a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.
Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.
The numerators of the main diagonal:
1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...
(thanks to Jean-François Alcover) are divisible by 2^n.
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