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A172194 Numerators of the inverse binomial transform of the sequence of fractions  A172030(n)/A172031(n). 0

%I

%S 0,1,1,2,2,19,19,23,23,131,131,808,808,4469,4469,24221,24221,-2797103,

%T -2797103,80009738,80009738,-930456539,-930456539,127441603151,

%U 127441603151,-6013673706973,-6013673706973,149990847412508,149990847412508

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

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

%C 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.

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

%C 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=0..n) B(i) = 1, 3/2, 4/3, 4/3, ... mentioned in A100649, then b(n)+c(n)=2. - _Paul Curtz_, Aug 04 2013.

%p 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:

%p L := [seq(c(n),n=0..30)] ; read("transforms") ; BINOMIALi(L) ; apply(numer,%) ; # _R. J. Mathar_, Dec 21 2010

%Y Cf. A100650 (denominators), A100649, A165142.

%K sign,frac

%O 0,4

%A _Paul Curtz_, Jan 29 2010

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Last modified October 15 04:33 EDT 2019. Contains 328026 sequences. (Running on oeis4.)