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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.
4
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
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
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Nov 05 2021
STATUS
approved