A349004 Decimal expansion of lim_{n->infinity} B(2*n, n)/n^(2*n), where B(n, x) is the n-th Bernoulli polynomial.
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, 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, 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
Offset: 0
Examples
0.313035285499331303636161246930847832912013941240452655543152967567084...
Links
- William Bell, Problem 4312, Crux Mathematicorum, Vol. 44, No. 2 (2018), pp. 69 and 71; Solution to Problem 4312, ibid., Vol. 45, No. 2 (2019), pp. 92-93.
- Eric Weisstein's World of Mathematics, Bernoulli Polynomial.
- Wikipedia, Bernoulli Polynomials.
- Index entries for transcendental numbers.
Programs
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Mathematica
$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}] RealDigits[2/(E^2 - 1), 10, 110][[1]]
Formula
Equals 2/(exp(2)-1).
From Peter Luschny, Nov 05 2021: (Start)
Equals lim_{n->oo} (1/n) * Sum_{k=0..n-1} B(2*n, 1 + k/n) by J. L. Raabe's multiplication theorem.
Equals -2 * lim_{n->oo} HurwitzZeta(1 - 2*n, n) * n^(1 - 2*n). (End)
Equals A073747 - 1. - Alois P. Heinz, Nov 05 2021
Equals Sum_{k>=1} tanh(1/2^k)/2^k (Bell, 2018). - Amiram Eldar, Apr 12 2022
Comments