A349004 - OEIS

%I #36 Apr 12 2022 03:31:49

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

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

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

%N Decimal expansion of lim_{n->infinity} B(2*n, n)/n^(2*n), where B(n, x) is the n-th Bernoulli polynomial.

%C Asymptotic expansion: B(2*n,n) / n^(2*n) ~ c0 + c1/n + c2/n^2 + ..., where

%C c0 = A349004

%C c1 = -0.11332842437985451266688985513574347679739396134203607414578687657...

%C c2 = -0.02939332883129837328682967905833985820907100422772261310141242364...

%C In general, for k>=1, B(k*n,n) / n^(k*n) ~ k/(exp(k) - 1).

%H William Bell, <a href="https://cms.math.ca/publications/crux/issue?volume=44&amp;issue=2">Problem 4312</a>, Crux Mathematicorum, Vol. 44, No. 2 (2018), pp. 69 and 71; <a href="https://cms.math.ca/publications/crux/issue?volume=45&amp;issue=2">Solution to Problem 4312</a>, ibid., Vol. 45, No. 2 (2019), pp. 92-93.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliPolynomial.html">Bernoulli Polynomial</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_polynomials">Bernoulli Polynomials</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals 2/(exp(2)-1).

%F From _Peter Luschny_, Nov 05 2021: (Start)

%F Equals lim_{n->oo} (1/n) * Sum_{k=0..n-1} B(2*n, 1 + k/n)) by J. L. Raabe's multiplication theorem.

%F Equals -2 * lim_{n->oo} HurwitzZeta(1 - 2*n, n) * n^(1 - 2*n). (End)

%F Equals A073747 - 1. - _Alois P. Heinz_, Nov 05 2021

%F Equals Sum_{k>=1} tanh(1/2^k)/2^k (Bell, 2018). - _Amiram Eldar_, Apr 12 2022

%e 0.313035285499331303636161246930847832912013941240452655543152967567084...

%t $MaxExtraPrecision = 1000; funs[n_] := BernoulliB[2 n, n]/n^(2 n); Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[1000/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 110]], {m, 10, 100, 10}]

%t RealDigits[2/(E^2 - 1), 10, 110][[1]]

%Y Cf. A053382, A053383, A073747, A349003.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Nov 05 2021