A165142 Numerators of a partial sum of 0, 1, 1/2, B_2, B_3, B_4,.., a modified Bernoulli sequence.

0, 0, 1, 3, 5, 5, 49, 49, 58, 58, 341, 341, 1963, 1963, 14479, 14479, 39236, 39236, -2286593, -2286593, 81626353, 81626353, -928516601, -928516601, 127463912438, 127463912438, -6013599342683, -6013599342683, 149990958958943

Offset: 0

Author

Paul Curtz, Sep 05 2009

Keywords

• sign
• frac

Comments

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.

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.

The first differences are f(n+1)-f(n) = b(n), by construction.

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).

Crossrefs

Cf. A100650 (denominators).

Programs

Maple

read("transforms") ; L := [0,0,1,1/2,seq(bernoulli(i),i=2..30)] ; PSUM(L) ; apply(numer,%) ; # R. J. Mathar, Dec 02 2010

Mathematica

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 *)