A393668 - OEIS

%I #21 Mar 02 2026 10:12:58

%S 1,3,0,1,8,4,6,3,9,8,6,0,3,7,1,2,6,7,7,7,7,0,4,3,3,6,6,3,0,0,7,8,9,5,

%T 3,3,0,1,3,1,0,1,7,5,1,4,2,7,9,4,4,5,1,8,3,9,9,6,7,6,4,9,6,4,2,1,2,8,

%U 1,4,8,6,2,2,6,0,9,9,3,4,1,4,1,2,9,1,1,7,1,9,0,1,8,8,1,8,9,7,1,6,6,2,5,9,2,5

%N Decimal expansion of log(sinh(Pi)/Pi).

%C Arrives as the k=2 case of the arctanh power sums 2*h(k) = Sum_{n>=2} arctanh(1/n^k).

%H Ryan Goulden, <a href="https://arxiv.org/abs/2602.06244">Closed-Form Evaluation of arctanh Power Sums via Infinite Products</a>, arXiv:2602.06244 [math.GM], 2026.

%F Equals log(sinh(Pi)/Pi) = log(A156648).

%F Equals 2 * Sum_{n>=2} arctanh(1/n^2).

%F Equals 2*Sum_{k>=0} (zeta(2*(2*k+1)) - 1) / (2*k+1).

%F Equals -2*A352527. - _Hugo Pfoertner_, Feb 24 2026

%e 1.301846398603712677770433663...

%t RealDigits[Log[Sinh[Pi]/Pi], 10, 106][[1]]

%o (PARI) log(sinh(Pi)/Pi)

%Y Cf. A000796, A090986, A156648, A352527.

%K nonn,cons

%O 1,2

%A _Ryan Goulden_, Feb 24 2026