A257106 -id:A257106 - cp's OEIS frontend

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

1, 0, -5, 3, -61, 5, -125, 7, -121, 9, -325, 11, -17071, 13, -35, 15, -7697, 17, 36685, 19, -177911, 21, 852995, 23, -236396851, 25, 8553025, 27, -23749473209, 29, 8615841061175, 31, -7709321049377, 33, 2577687858265, 35, -26315271553088022793, 37

Offset: 0

Paul Curtz, May 13 2015

sign

Difference table of 1, 1, 1/6, 0, -1/30, ... :
1,            1,    1/6,      0,  -1/30,     0,  1/42, 0, ...
0,         -5/6,   -1/6,  -1/30,   1/30,  1/42, -1/42, ...
-5/6,       2/3,   2/15,   1/15, -1/105, -1/21, ...
3/2,      -8/15,  -1/15, -8/105, -4/105, ...
-61/30,    7/15, -1/105,  4/105, ...
5/2,     -10/21,   1/21, ...
-125/42,  11/21, ...
7/2, ...
etc.
The inverse binomial transform is the first column. a(n) is the n-th term of the numerators. See A027641(n+1).
Denominators: A176591.
Is a(4n+2) a multiple of 5? This is true, at least up to 4n+2 = 998. - Jean-François Alcover, Jul 02 2015

			By the first formula: numerators of 1-0=1, -1/2+1/2=0, 1/6-1=-5/6, 0+3/2=3/2,....

Colin Barker, Table of n, a(n) for n = 0..629

Cf. A257106, A027641/A027642, A164555/A027642, A176327/A027642, A176328/A176591, A026741.

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 *)

firstdiff(s) = my(t=vector(#s-1)); for(i=2, #s, t[i-1]=s[i]-s[i-1]); t
a257935(k) = {
  my(s=[], b = concat([1,1], vector(k, n, n++; bernfrac(n))));
  until(#b<2,
    s = concat(s, numerator(b[1]));
    b = firstdiff(b)
  );
  s
}
a257935(50) \\ Colin Barker, May 13 2015

a(n) = numerators of A027641(n)/A027642(n) - (-1)^n*n/2.
a(n) = (A176328(n) - (-1)^n*n)*A176591(n).
a(n) = 2*A027641(n)*A176591(n)/A027642(n) - A176328(n).

More terms from Colin Barker, May 13 2015

cp's OEIS Frontend

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

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