This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A172194 #22 Feb 07 2025 08:01:04
%S A172194 0,1,1,2,2,19,19,23,23,131,131,808,808,4469,4469,24221,24221,-2797103,
%T A172194 -2797103,80009738,80009738,-930456539,-930456539,127441603151,
%U A172194 127441603151,-6013673706973,-6013673706973,149990847412508,149990847412508
%N A172194 Numerators of the inverse binomial transform of the sequence of fractions A172030(n)/A172031(n).
%C A172194 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 A172194 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 A172194 Also the numerators of the partial sums of the Bernoulli Numbers, Sum_{i=0..n} B(i). - _Paul Curtz_, Aug 02 2013
%C A172194 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.
%p A172194 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 A172194 L := [seq(c(n),n=0..30)] ; read("transforms") ; BINOMIALi(L) ; apply(numer,%) ; # _R. J. Mathar_, Dec 21 2010
%Y A172194 Cf. A100650 (denominators), A100649, A165142.
%K A172194 sign,frac
%O A172194 0,4
%A A172194 _Paul Curtz_, Jan 29 2010