OFFSET
0,10
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,... .
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).
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.
FORMULA
CROSSREFS
KEYWORD
sign
AUTHOR
Paul Curtz, May 09 2014
STATUS
approved