<s>
In	O
mathematics	O
,	O
the	O
boustrophedon	B-Algorithm
transform	I-Algorithm
is	O
a	O
procedure	O
which	O
maps	O
one	O
sequence	O
to	O
another	O
.	O
</s>
<s>
The	O
boustrophedon	B-Algorithm
transform	I-Algorithm
is	O
a	O
numerical	O
,	O
sequence-generating	O
transformation	O
,	O
which	O
is	O
determined	O
by	O
an	O
"	O
addition	O
"	O
operation	O
.	O
</s>
<s>
Generally	O
speaking	O
,	O
given	O
a	O
sequence	O
:	O
,	O
the	O
boustrophedon	B-Algorithm
transform	I-Algorithm
yields	O
another	O
sequence	O
:	O
,	O
where	O
is	O
likely	O
defined	O
equivalent	O
to	O
.	O
</s>
<s>
In	O
the	O
case	O
a0	O
=	O
1	O
,	O
an	O
=	O
0	O
(	O
n	O
0	O
)	O
,	O
the	O
resulting	O
triangle	O
is	O
called	O
the	O
Seidel	O
–	O
Entringer	O
–	O
Arnold	O
Triangle	O
and	O
the	O
numbers	O
are	O
called	O
Entringer	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
In	O
this	O
case	O
the	O
numbers	O
in	O
the	O
transformed	O
sequence	O
bn	O
are	O
called	O
the	O
Euler	O
up/down	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
These	O
enumerate	O
the	O
number	O
of	O
alternating	B-Algorithm
permutations	I-Algorithm
on	O
n	O
letters	O
and	O
are	O
related	O
to	O
the	O
Euler	O
numbers	O
and	O
the	O
Bernoulli	O
numbers	O
.	O
</s>
<s>
Building	O
from	O
the	O
geometric	O
design	O
of	O
the	O
boustrophedon	B-Algorithm
transform	I-Algorithm
,	O
algebraic	O
definitions	O
of	O
the	O
relationship	O
from	O
input	O
values	O
(	O
)	O
to	O
output	O
values	O
(	O
)	O
can	O
be	O
defined	O
for	O
different	O
algebras	O
(	O
"	O
numeric	O
domains	O
"	O
)	O
.	O
</s>
<s>
where	O
is	O
the	O
sequence	O
of	O
"	O
up/down	O
"	O
numbers	O
—	O
also	O
known	O
as	O
secant	B-Algorithm
or	O
tangent	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
exponential	O
generating	O
function	O
of	O
the	O
unit	O
sequence	O
is1	O
,	O
so	O
that	O
of	O
the	O
up/down	B-Algorithm
numbers	I-Algorithm
is	O
secx+tanx	O
.	O
</s>
