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

Author

Vaclav Kotesovec, Nov 05 2021

Keywords

• nonn
• cons

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)).

Examples

			0.238405844044235111880541717395206409587231402742063448403189499878046...
		

Links

• Eric Weisstein's World of Mathematics, Euler Polynomial.

Crossrefs

Cf. A004174, A292782, A349004.

Programs

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]]

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).