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A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

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