A172194 Numerators of the inverse binomial transform of the sequence of fractions A172030(n)/A172031(n).

0, 1, 1, 2, 2, 19, 19, 23, 23, 131, 131, 808, 808, 4469, 4469, 24221, 24221, -2797103, -2797103, 80009738, 80009738, -930456539, -930456539, 127441603151, 127441603151, -6013673706973, -6013673706973, 149990847412508, 149990847412508

Offset: 0

Author

Paul Curtz, Jan 29 2010

Keywords

• sign
• frac

Comments

The original sequence starts 0, 1, 5/2, 31/6, 31/3, 619/30, 619/15, 5779/70, 5779/35, 69341/210, 69341/105, ...

The inverse binomial transform yields 0, 1, 1/2, 2/3, 2/3, 19/30, 19/30, 23/35, 23/35, 131/210, 131/210, 808/1155, ... with numerators defining the sequence.

Also the numerators of the partial sums of the Bernoulli Numbers, Sum_{i=0..n} B(i). - Paul Curtz, Aug 02 2013

If we consider this sequence of partial sums b(n) := Sum_{i=0..n} B(i) = 1, 1/2, 2/3, 2/3, ... and also the sequence c(n) := 1 - Sum_{i=1..n} B(i) = 1, 3/2, 4/3, 4/3, ... mentioned in A100649, then b(n)+c(n)=2. - Paul Curtz, Aug 04 2013.

Crossrefs

Cf. A100650 (denominators), A100649, A165142.

Programs

Maple

c := proc(n) option remember; if n <=1 then n; elif n = 2 then 2*procname(n-1)-bernoulli(n-1) ; else 2*procname(n-1)+bernoulli(n-1) ; end if; end proc:
L := [seq(c(n),n=0..30)] ; read("transforms") ; BINOMIALi(L) ; apply(numer,%) ; # R. J. Mathar, Dec 21 2010