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%I A165142 #9 Aug 09 2012 04:44:13
%S A165142 0,0,1,3,5,5,49,49,58,58,341,341,1963,1963,14479,14479,39236,39236,
%T A165142 -2286593,-2286593,81626353,81626353,-928516601,-928516601,
%U A165142 127463912438,127463912438,-6013599342683,-6013599342683,149990958958943
%N A165142 Numerators of a partial sum of 0, 1, 1/2, B_2, B_3, B_4,.., a modified Bernoulli sequence.
%C A165142 A modified list of Bernoulli numbers starts b(n) = 0, 1, 1/2, 1/6, 0, -1/30, 0, 1/42,..., n>=0, which is the standard Bernoulli sequence A027641(.)/A027642(.), prefixed with a zero and sign flipped at B_1 = -1/2.
%C A165142 Building partial sums of b(n) yields f(n) = 0, 0, 1, 3/2, 5/3, 5/3, 49/30, 49/30, 58/35, 58/35, 341/210, 341/210, 1963/1155,...., n>=0. The numerators of f(n) define the current sequence; denominators are found by prefixing A100650 with two 1's.
%C A165142 The first differences are f(n+1)-f(n) = b(n), by construction.
%C A165142 The inverse binomial transform of f(n) is (-1)^n*f(n); the inverse binomial transform of b(n) is 0, 1, -3/2, 5/3, -5/3, 49/30, -49/30,... an alternating sign variant of a shifted f(n).
%p A165142 read("transforms") ; L := [0,0,1,1/2,seq(bernoulli(i),i=2..30)] ; PSUM(L) ; apply(numer,%) ; # _R. J. Mathar_, Dec 02 2010
%t A165142 b[n_] := BernoulliB[n-1]; b[0]=0; b[1]=1; b[2]=1/2; Join[{0}, Accumulate[ Table[b[n], {n, 0, 27}]] // Numerator] (* Jean-François Alcover_, Aug 09 2012 *)
%Y A165142 Cf. A100650 (denominators).
%K A165142 sign,frac
%O A165142 0,4
%A A165142 _Paul Curtz_, Sep 05 2009

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