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A238398 Numerators of inverse binomial transform of A198631(n)/A006519(n+1) with -1 instead of A198631(1)=1.

%I A238398 #16 Mar 05 2014 11:09:30
%S A238398 1,-3,2,-11,4,-11,6,-39,8,-49,10,647,12,-5487,14,929329,16,-3202325,
%T A238398 18,221930505,20,-4722116563,22,968383680643,24,-14717667114201,26,
%U A238398 2093660879252563,28,-86125672563201239,30,129848163681107300961,32
%N A238398 Numerators of inverse binomial transform of A198631(n)/A006519(n+1) with -1 instead of A198631(1)=1.
%C A238398 From modified fractional Euler numbers.
%C A238398 Inverse binomial transform:
%C A238398 1, -3/2, 2, -11/4, 4, -11/2, 6, -39/8, 8, -49/2, 10, 647/4, 12, -5487/2,... = a(n)/b(n).  b(2n) = A004277(n).
%C A238398 Difference table of c(n) = 1, -1/2, 0, -1/4,... :
%C A238398 1,    -1/2,    0,  -1/4,     0,    1/2,      0,...
%C A238398 -3/2,  1/2, -1/4,   1/4,   1/2,   -1/2,  -17/8,...
%C A238398 2,    -3/4,  1/2,   1/4,    -1,  -13/8,   17/4,...
%C A238398 -11/4, 5/4, -1/4,  -5/4,  -5/8,   47/8,   73/8,...
%C A238398 4,    -3/2,   -1,   5/8,  13/2,   13/4, -107/2,...
%C A238398 -11/2, 1/2, 13/8,  47/8, -13/4, -227/4, -227/8,
%C A238398 6,     9/8, 17/4, -73/8, -107/2, 227/8, 2957/4,...
%C A238398 etc.
%C A238398 c(n) + a(n)/b(n) = 2, -2, 2, -3, 4, -5, 6, -7, 8, -9,... = A233583(n+1) signed. (a(n) discovered in 2013)
%t A238398 max = 40;(* b = A198631 *) b[0] = 1; b[1] = -1; b[n_] := Numerator[EulerE[n, 1]/(2^n-1)]; bb = Table[b[n]/2^IntegerExponent[n+1, 2], {n, 0, max}]; a[n_] := Differences[bb, n] // First // Numerator ; Table[a[n], {n, 0, max}]
%Y A238398 Cf. A235774.
%K A238398 sign
%O A238398 0,2
%A A238398 _Paul Curtz_, Feb 26 2014