A349003 Decimal expansion of lim_{n->infinity} E(2*n, n)/n^(2*n), where E(n, x) is the n-th Euler polynomial.

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

Offset

0,1

Comments

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

c0 = A349003

c1 = -0.15992500211230612504712294232596098830480284076519978623574964079...

c2 = -0.07258631854606119935476518617230181507488028047324715883939525404...

In general, for k>=1, E(k*n,n) / n^(k*n) ~ 2/(1 + exp(k)).

Links

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

Eric Weisstein's World of Mathematics, Euler Polynomial.

Formula

Equals 2/(1 + exp(2)).

Equals lim_{n->infinity} (HurwitzZeta(-2*n, n/2) - HurwitzZeta(-2*n, (n+1)/2)) * 2^(2*n+1) / n^(2*n).

Example

0.238405844044235111880541717395206409587231402742063448403189499878046...

Mathematica

$MaxExtraPrecision = 1000; funs[n_] := EulerE[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/(1 + E^2), 10, 110][[1]]

Crossrefs

Cf. A004174, A292782, A349004.

Sequence in context: A155994 A011162 A293273 * A079555 A100870 A210688

Adjacent sequences: A349000 A349001 A349002 * A349004 A349005 A349006

Keyword

nonn,cons

Author

Vaclav Kotesovec, Nov 05 2021

Status

approved