The Akiyama-Tanigawa algorithm is a triangular recurrence for computing Bernoulli numbers and transforming sequences (Kaneko 2000, Akiyama and Tanigawa 2001). Starting with
 |
(1)
|
define subsequent rows by
 |
(2)
|
for nonnegative integers 
and 
.
The leading element of row 
is
 |
(3)
|
where 
is a Bernoulli polynomial and 
uses the convention 
. Thus the algorithm gives 
and agrees with 
at every other index (Kaneko 2000, Chen 2001).
More generally, an arbitrary initial row 
gives
 |
(4)
|
where 
is a Stirling number of the second
kind. If the initial row has ordinary generating
function 
, then the leading column
has exponential generating function
 |
(5)
|
These identities hold as formal power series (Kaneko 2000).
Chen's (2001) variation uses the same initial row and the recurrence
 |
(6)
|
Its leading column satisfies
 |  |  |
(7)
|
 |  |  |
(8)
|
In particular, 
gives the conventional Bernoulli
numbers 
. The choice 
gives the Euler polynomial
values 
under the original recurrence and 
under the variation. The choice 
gives the Bell numbers
with indices 
and 
,
respectively (Chen 2001).
The initial row with ordinary generating function
 |
(9)
|
gives the signed Euler numbers 
under the original recurrence. Under the variation, the
leading column has exponential generating
function 
. Writing its entries as 
, the initial value is 
and
 |  |  |
(10)
|
 |  |  |
(11)
|
for 
, where 
is a tangent number in
the positive convention 
, 
, 
,
... (Chen 2001).
A two-parameter generalization replaces the recurrence by
 |
(12)
|
For nonzero 
and 
, its leading-column exponential
generating function is
 |
(13)
|
(Rahmani 2013).
