OFFSET
0,4
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).
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 *)
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Paul Curtz, Sep 05 2009
STATUS
approved