Somos' quadratic recurrence constant

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).