In mathematics, **Somos' quadratic recurrence constant**, named after Michael Somos, is the number

σ
=
1
2
3
⋯
=
1
1
/
2
2
1
/
4
3
1
/
8
⋯
.
This can be easily re-written into the far more quickly converging product representation

σ
=
σ
2
/
σ
=
(
2
1
)
1
/
2
(
3
2
)
1
/
4
(
4
3
)
1
/
8
(
5
4
)
1
/
16
⋯
.
The constant σ arises when studying the asymptotic behaviour of the sequence

g
0
=
1
;
g
n
=
n
g
n
−
1
2
,
n
>
1
,
with first few terms 1, 1, 2, 12, 576, 1658880 ... (sequence A052129 in the OEIS). This sequence can be shown to have asymptotic behaviour as follows:

g
n
∼
σ
2
n
n
+
2
+
O
(
1
n
)
.
Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent:

ln
σ
=
−
1
2
∂
Φ
∂
s
(
1
2
,
0
,
1
)
where ln is the natural logarithm and
Φ
(*z*, *s*, *q*) is the Lerch transcendent.

Using series acceleration it is the sum of the n-th differences of ln(k) at k=1 as given by:

ln
σ
=
∑
n
=
1
∞
∑
k
=
0
n
(
−
1
)
n
−
k
(
n
k
)
ln
(
k
+
1
)
.
Finally,

σ
=
1.661687949633594121296
…
(sequence

A112302 in the OEIS).