A241519 Denominators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

1, 2, 2, 12, 3, 15, 60, 840, 105, 630, 630, 13860, 6930, 180180, 360360, 144144, 9009, 306306, 306306, 11639628, 14549535, 14549535, 58198140, 2677114440, 334639305, 3346393050

Offset: 0

Author

Paul Curtz, Apr 24 2014

Keywords

• nonn

Comments

Generally, 2*b(n) = b(n-1) + f(n). See, for f(n)=n, A000337(n)/2^n.

a(0)=1. b(n) is mentioned in A241269.

Difference table of b(n):

0, 1/2, 1/2, 5/12, 1/3, 4/15, ...

1/2, 0, -1/12, -1/12, -1/15, -1/20, ...

-1/2, -1/12, 0, 1/60, 1/60, 11/840, ...

5/12, 1/12, 1/60, 0, -1/280, -1/280, ...

etc.

b(n) is mentioned in A241269 as an autosequence of the first kind.

The denominators of the first two upper diagonals are the positive Apéry numbers, A005430(n+1). Compare to the array in A003506.

Numerators: 0, 1, 1, 5, 1, 4, 13, 151, 16, 83, 73, 1433, 647, 15341, ... .

Examples

			0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ...
b(1) = (0+1)/2, hence a(1)=2.
b(2) = (1/2+1/2)/2 = 1/2, hence a(2)=2.
b(3) = (1/2+1/3)/2 = 5/12, hence a(3)=12.
		

Crossrefs

Cf. A086466.

Cf. A242376 (numerators).

Programs

Mathematica

b[0] = 0; b[n_] := b[n] = 1/2*(b[n-1] + 1/n); Table[b[n] // Denominator, {n, 0, 25}] (* Jean-François Alcover, Apr 25 2014 *)
Table[-Re[LerchPhi[2, 1, n + 1]], {n, 0, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
-Re[LerchPhi[2, 1, Range[20]]] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
RecurrenceTable[{b[n] == b[n - 1]/2 + 1/(2 n), b[0] == 0}, b[n], {n, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)

Formula

b(n) = -Re(Phi(2, 1, n + 1)) where Phi denotes the Lerch transcendent. - Eric W. Weisstein, Dec 11 2017

Extensions

Extension, after a(13), from Jean-François Alcover, Apr 24 2014