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A165142 Numerators of a partial sum of 0, 1, 1/2, B_2, B_3, B_4,.., a modified Bernoulli sequence. 6
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 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=0..28.

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-Fran├žois Alcover_, Aug 09 2012 *)

CROSSREFS

Cf. A100650 (denominators).

Sequence in context: A072624 A147976 A019247 * A231809 A186969 A111950

Adjacent sequences:  A165139 A165140 A165141 * A165143 A165144 A165145

KEYWORD

sign,frac

AUTHOR

Paul Curtz, Sep 05 2009

STATUS

approved

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Last modified November 19 12:57 EST 2018. Contains 317351 sequences. (Running on oeis4.)