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A349003 Decimal expansion of lim_{n->infinity} E(2*n, n)/n^(2*n), where E(n, x) is the n-th Euler polynomial. 3
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified January 26 07:32 EST 2022. Contains 350577 sequences. (Running on oeis4.)