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

1, -3, 2, -11, 4, -11, 6, -39, 8, -49, 10, 647, 12, -5487, 14, 929329, 16, -3202325, 18, 221930505, 20, -4722116563, 22, 968383680643, 24, -14717667114201, 26, 2093660879252563, 28, -86125672563201239, 30, 129848163681107300961, 32

Offset: 0

Author

Paul Curtz, Feb 26 2014

Keywords

• sign

Comments

From modified fractional Euler numbers.

Inverse binomial transform:

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).

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

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

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

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

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

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

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

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

etc.

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)

Crossrefs

Cf. A235774.

Programs

Mathematica

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}]