A242246 Numerators of n*A164555(n-1)/A027642(n-1).

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615641276005, 0, -84802531453387, 0, 90219075042845, 0

Offset: 0

Author

Paul Curtz, May 09 2014

Keywords

• sign

Comments

First multiplied shifted (second) Bernoulli numbers.

A164555(n-1)/A027642(n-1) = 0 followed by (A164555(n)/A027642(n)=1, 1/2, 1/6,...) = f(n) = 0, 1, 1/2, 1/6, 0,... .

f(n+1) - f(n) = A051716(n)/A051717(n).

Generally we consider a transform applied to the autosequences of first or second kind. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. It is of the first kind if the main diagonal is A000004=0's. It is of the second kind if the main diagonal is the first upper diagonal multiplied by 2. A000045(n) is an autosequence of the first kind. A164555(n)/A027642(n) is an autosequence of the second kind. See A190339 (and A241269).

Here we apply the transform to the Bernoulli numbers A164555(n)/A027642(n).

We take n*(0 followed by A164555(n)/A027642(n)).

Hence the autosequence of first kind

TB1(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, O, -691/210,.. .

a(n) are the numerators.

The first seven rows of the differencece table of TB1(n) are

0, 1, 1, 1/2, 0, - 1/6, 0, 1/6,...

1, 0, -1/2, -1/2, -1/6, 1/6, 1/6, -1/6,... =A140351(n+1)/b(n+1)

-1, -1/2, 0, 1/3, 1/3, 0, -1/3, -2/15,...

1/2, 1/2, 1/3, 0, -1/3, -1/3, 1/5, 11/15,...

0, -1/6, -1/3, -1/3, 0, 8/15, 8/15, -4/5,...

-1/6, -1/6, 0, 1/3, 8/15, 0, -4/3, -4/3,...

0, 1/6, 1/3, 1/5, -8/15, -4/3, 0, 512/105,... .

First and second upper diagonals: 1, -1/2, 1/3, -1/3, 8/15, -4/3, 512/105,... .

Sum of the antidiagonals:

0, 1, 1, 0, -1/2, 0, 1/2, 0, -5/6, 0, 13/6, 0, -49/6, 0,... .

(Note that the same transform applied to the second fractional Euler numbers A198631(n)/A006519(n+1) yields the Genocchi numbers -A226158(n)).

This transform can be continued:

TB2(n) = n*(0 followed by TB1(n)) =

0, 0, 2, 3, 2, 0, -1, 0, 4/3, 0, -3, 0, 10, 0, -691/15, 0, 280, 0,...

is an autosequence of second kind.

TB3(n) = 0, 0, 0, 6, 12, 10, 0, -7, 0, 12, 0, -33, 0, 130, 0, 691, 0,...

is apparently an integer autosequence of the first kind.

Crossrefs

Cf. A199969 (autosequence).

Formula

a(n) = 0 followed by (A050925(n) = 1, -1, 1, 0,... ) with 1 instead of -1.

a(2n) = A063524(n). a(2n+1) = A002427(n).