A244605 Numerators of the Akiyama-Tanigawa transform applied to 1/(n+1) with -1/2 instead of 1/2.

1, 3, 19, 7, 449, 31, 2647, 127, 7649, 511, 67523, 2047, 11178659, 8191, 98305, 32767, 33419233, 131071, 209233981, 524287, 345855139, 2097151, 579668327, 8388607, 45565432859, 33554431, 411206281, 134217727, 209789384821, 536870911, 23993971665011, 2147483647, -5518887720767

Offset: 0

Author

Paul Curtz, Jul 01 2014

Keywords

• sign
• frac

Comments

The autosequence of the second kind A164555(n)/A027642(n) = 1, 1/2, 1/6, 0, -1/30, 0, ... (the second Bernoulli numbers) is the binomial transform of A027641(n)/A027642(n) = 1, -1/2, 1/6, 0, -1/30, 0, ... (the first Bernoulli numbers). Hence the name.

The Akiyama-Tanigawa transform applied to 1, -1/2, 1/3, 1/4, 1/5, 1/6, ... is:

1, -1/2, 1/3, 1/4, 1/5, ...

3/2, -5/3, 1/4, 1/5, 1/6, ...

19/6, -23/6, 3/20, 2/15, 5/42, ...

7, -239/30, 1/20, 2/35, 5/84, ... .

The first column is a(n)/b(n) = 1, 3/2, 19/6, 7, 449/30, 31, 2647/42, 127, 7649/30, 511, 67523/66, 2047, ..., where the denominators are b(n) = A027642(n).

By the formula below, the Bernoulli numbers are linked to the Mersenne numbers A000225 (2^n-1).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A190339, A051715, A027641, A164555, A027642, A000225.

Programs

Mathematica

a[n_] := BernoulliB[n]+2^n-1 // Numerator; a[1] = 3; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jul 25 2014 *)

PARI

a(n) = my(b = numerator(bernfrac(n))/denominator(bernfrac(n))); if (n == 1, numerator(- b + 2^n - 1), numerator(b + 2^n - 1)); \\ Michel Marcus, Jul 18 2014

PARI

{a(n) = if( n<0, 0, 2*(n==1) + numerator( bernfrac(n) + 2^n - 1))}; /* Michael Somos, Aug 05 2014 */

Formula

a(n) = numerator of A164555(n)/A027642(n) + A000225(n).

Extensions

a(12)-a(32) from Jean-François Alcover, Jul 01 2014