A244605 - OEIS

%I #29 Aug 07 2018 00:42:02

%S 1,3,19,7,449,31,2647,127,7649,511,67523,2047,11178659,8191,98305,

%T 32767,33419233,131071,209233981,524287,345855139,2097151,579668327,

%U 8388607,45565432859,33554431,411206281,134217727,209789384821,536870911,23993971665011,2147483647,-5518887720767

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

%C 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.

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

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

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

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

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

%C 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).

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

%H G. C. Greubel, <a href="/A244605/b244605.txt">Table of n, a(n) for n = 0..500</a>

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

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

%o (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

%o (PARI)  {a(n) = if( n<0, 0, 2*(n==1) + numerator( bernfrac(n) + 2^n - 1))}; /* _Michael Somos_, Aug 05 2014 */

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

%K sign,frac

%O 0,2

%A _Paul Curtz_, Jul 01 2014

%E a(12)-a(32) from _Jean-François Alcover_, Jul 01 2014