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

0,3

Comments

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

Links

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

Formula

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).

Example

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

Mathematica

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 *)

Prog

(PARI)
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

Crossrefs

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

Sequence in context: A027858 A181755 A007299 * A109254 A258091 A370551

Adjacent sequences: A257932 A257933 A257934 * A257936 A257937 A257938

Keyword

sign

Author

Paul Curtz, May 13 2015

Extensions

More terms from Colin Barker, May 13 2015

Status

approved