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

%I #16 Mar 05 2014 11:09:30

%S 1,-3,2,-11,4,-11,6,-39,8,-49,10,647,12,-5487,14,929329,16,-3202325,

%T 18,221930505,20,-4722116563,22,968383680643,24,-14717667114201,26,

%U 2093660879252563,28,-86125672563201239,30,129848163681107300961,32

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

%C From modified fractional Euler numbers.

%C Inverse binomial transform:

%C 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 Difference table of c(n) = 1, -1/2, 0, -1/4,... :

%C 1, -1/2, 0, -1/4, 0, 1/2, 0,...

%C -3/2, 1/2, -1/4, 1/4, 1/2, -1/2, -17/8,...

%C 2, -3/4, 1/2, 1/4, -1, -13/8, 17/4,...

%C -11/4, 5/4, -1/4, -5/4, -5/8, 47/8, 73/8,...

%C 4, -3/2, -1, 5/8, 13/2, 13/4, -107/2,...

%C -11/2, 1/2, 13/8, 47/8, -13/4, -227/4, -227/8,

%C 6, 9/8, 17/4, -73/8, -107/2, 227/8, 2957/4,...

%C etc.

%C 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 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 Cf. A235774.

%K sign

%O 0,2

%A _Paul Curtz_, Feb 26 2014