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%I #21 Nov 06 2021 03:41:31
%S 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,
%T 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,
%U 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
%N Decimal expansion of lim_{n->infinity} E(2*n, n)/n^(2*n), where E(n, x) is the n-th Euler polynomial.
%C Asymptotic expansion: E(2*n,n) / n^(2*n) ~ c0 + c1/n + c2/n^2 + ..., where
%C c0 = A349003
%C c1 = -0.15992500211230612504712294232596098830480284076519978623574964079...
%C c2 = -0.07258631854606119935476518617230181507488028047324715883939525404...
%C In general, for k>=1, E(k*n,n) / n^(k*n) ~ 2/(1 + exp(k)).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerPolynomial.html">Euler Polynomial</a>.
%F Equals 2/(1 + exp(2)).
%F Equals lim_{n->infinity} (HurwitzZeta(-2*n, n/2) - HurwitzZeta(-2*n, (n+1)/2)) * 2^(2*n+1) / n^(2*n).
%e 0.238405844044235111880541717395206409587231402742063448403189499878046...
%t $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}]
%t RealDigits[2/(1 + E^2), 10, 110][[1]]
%Y Cf. A004174, A292782, A349004.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, Nov 05 2021
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