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A257935 Numerators of the inverse binomial transform of the Bernoulli numbers with B(1)=1.

%I A257935 #26 Jul 02 2015 10:17:23
%S A257935 1,0,-5,3,-61,5,-125,7,-121,9,-325,11,-17071,13,-35,15,-7697,17,36685,
%T A257935 19,-177911,21,852995,23,-236396851,25,8553025,27,-23749473209,29,
%U A257935 8615841061175,31,-7709321049377,33,2577687858265,35,-26315271553088022793,37
%N A257935 Numerators of the inverse binomial transform of the Bernoulli numbers with B(1)=1.
%C A257935 Difference table of 1, 1, 1/6, 0, -1/30, ... :
%C A257935 1,            1,    1/6,      0,  -1/30,     0,  1/42, 0, ...
%C A257935 0,         -5/6,   -1/6,  -1/30,   1/30,  1/42, -1/42, ...
%C A257935 -5/6,       2/3,   2/15,   1/15, -1/105, -1/21, ...
%C A257935 3/2,      -8/15,  -1/15, -8/105, -4/105, ...
%C A257935 -61/30,    7/15, -1/105,  4/105, ...
%C A257935 5/2,     -10/21,   1/21, ...
%C A257935 -125/42,  11/21, ...
%C A257935 7/2, ...
%C A257935 etc.
%C A257935 The inverse binomial transform is the first column. a(n) is the n-th term of the numerators. See A027641(n+1).
%C A257935 Denominators: A176591.
%C A257935 Is a(4n+2) a multiple of 5? This is true, at least up to 4n+2 = 998. - _Jean-François Alcover_, Jul 02 2015
%H A257935 Colin Barker, <a href="/A257935/b257935.txt">Table of n, a(n) for n = 0..629</a>
%F A257935 a(n) = numerators of A027641(n)/A027642(n) - (-1)^n*n/2.
%F A257935 a(n) = (A176328(n) - (-1)^n*n)*A176591(n).
%F A257935 a(n) = 2*A027641(n)*A176591(n)/A027642(n) - A176328(n).
%e A257935 By the first formula: numerators of 1-0=1, -1/2+1/2=0, 1/6-1=-5/6, 0+3/2=3/2,....
%t A257935 max = 40; B[1] = 1; B[n_] := BernoulliB[n]; BB = Array[B, max, 0]; a[n_] := Differences[BB, n] // First // Numerator; Table[a[n], {n, 0, max-1}] (* _Jean-François Alcover_, May 20 2015 *)
%o A257935 (PARI)
%o A257935 firstdiff(s) = my(t=vector(#s-1)); for(i=2, #s, t[i-1]=s[i]-s[i-1]); t
%o A257935 a257935(k) = {
%o A257935   my(s=[], b = concat([1,1], vector(k, n, n++; bernfrac(n))));
%o A257935   until(#b<2,
%o A257935     s = concat(s, numerator(b[1]));
%o A257935     b = firstdiff(b)
%o A257935   );
%o A257935   s
%o A257935 }
%o A257935 a257935(50) \\ _Colin Barker_, May 13 2015
%Y A257935 Cf. A257106, A027641/A027642, A164555/A027642, A176327/A027642, A176328/A176591, A026741.
%K A257935 sign
%O A257935 0,3
%A A257935 _Paul Curtz_, May 13 2015
%E A257935 More terms from _Colin Barker_, May 13 2015