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%I #12 Jun 08 2019 08:09:09
%S 1,3,33,1011,65985,7536099,1329205857,334169853267,113370124235649,
%T 49880529542872515,27614111852126579361,18782012442066306225843,
%U 15394836674855296870428993,14965462261283347594195897251,17023467576167762236198869304545,22400927665017118737825435362462739
%N Expansion of exp( Sum_{n >= 1} A000281(n)*x^n/n ).
%C A000281(n) =(-1)^n*4^(2*n)*E(2*n,1/4), where E(n,x) denotes the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255881 (k = 2), A255882(k = 3) and A255884 (k = 6).
%H G. C. Greubel, <a href="/A255883/b255883.txt">Table of n, a(n) for n = 0..200</a>
%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/EulerPolynomial.html">Euler Polynomial</a>
%F O.g.f.: exp( 3*x + 57*x^2/2 + 2763*x^3/3 + 250737*x^4/4 + ... ) = 1 + 3*x + 33*x^2 + 1011*x^3 + 65985*x^4 + ....
%F a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*4^(2*k)*E(2*k,1/4)*a(n-k).
%F a(n) ~ 2^(6*n + 5/2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - _Vaclav Kotesovec_, Jun 08 2019
%p #A255883
%p k := 4:
%p exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);
%t A000281:= With[{nn = 200}, Take[CoefficientList[Series[Cos[x]/Cos[2 x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A000281[[k + 1]]*x^(k)/(k), {k, 1, 85}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Aug 26 2018 *)
%Y Cf. A000281, A188514, A255881, A255882, A255884
%K nonn,easy
%O 0,2
%A _Peter Bala_, Mar 09 2015
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