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%I #24 Apr 09 2023 07:54:21
%S 1,1,-1,-1,7,3,-31,-17,127,155,-2555,-2073,1414477,38227,-57337,
%T -929569,118518239,28820619,-5749691557,-1109652905,91546277357,
%U 51943281731,-1792042792463,-2905151042481,1982765468311237,191329672483963,-286994504449393,-14655626154768697,3187598676787461083,1291885088448017715,-4625594554880206790555
%N a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).
%C a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.
%C The numbers are
%C BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..
%C Difference table:
%C 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42,...
%C -1/2, -2/3, -1/3, 11/15, 19/15, -47/21, -163/21,...
%C -1/6, 1/3, 16/15, 8/15, -368/105, -116/21, 2152/105,...
%C 1/2, 11/15, -8/15, -424/105, -212/105, 2732/105, 4204/105,...
%C 7/30, -19/15, -368/195, 212/105, 2944/105, 1472/105,...
%C -3/2, -47/21, 116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .
%C a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.
%C a(n) is divisible by A051716(n+1).
%C Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.
%C The numerators of the main diagonal:
%C 1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...
%C (thanks to Jean-François Alcover) are divisible by 2^n.
%F c(n)=(0 followed by -A036968(n+1)) = 0, 1, 0, -1, 0, 3,... .
%F a(n) = A157779(n) + c(n).
%p A225825 := proc(n)
%p local nhalf ;
%p nhalf := floor(n/2) ;
%p if type(n,'even') then
%p A001896(nhalf) ;
%p else
%p (-1)^nhalf*A110501(nhalf+1) ;
%p end if;
%p end proc; # _R. J. Mathar_, Oct 28 2013
%t a[0] = 1; a[n_] := Numerator[BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 01 2013 *)
%Y Cf. A083420.
%K sign
%O 0,5
%A _Paul Curtz_, Jul 30 2013
%E More terms from _Jean-François Alcover_, Aug 01 2013
%E Definition corrected by _R. J. Mathar_, Oct 28 2013
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