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

0,1

Comments

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

c0 = A349004

c1 = -0.11332842437985451266688985513574347679739396134203607414578687657...

c2 = -0.02939332883129837328682967905833985820907100422772261310141242364...

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

Links

Table of n, a(n) for n=0..105.

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.

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

Example

0.313035285499331303636161246930847832912013941240452655543152967567084...

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]]

Crossrefs

Cf. A053382, A053383, A073747, A349003.

Sequence in context: A112298 A011430 A073747 * A127549 A191009 A082648

Adjacent sequences: A349001 A349002 A349003 * A349005 A349006 A349007

Keyword

nonn,cons

Author

Vaclav Kotesovec, Nov 05 2021

Status

approved