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A255882 Expansion of exp( Sum_{n >= 1} A210657(n)*(-x)^n/n ).

%I A255882 #15 Feb 16 2025 08:33:25
%S A255882 1,2,13,224,8170,522716,51749722,7309866728,1394040714169,
%T A255882 344865267322010,107361980072755261,41067497940750566312,
%U A255882 18931745446455458282248,10350955324610065848650384,6622526747212249020075069880,4901565185965701578921602882976
%N A255882 Expansion of exp( Sum_{n >= 1} A210657(n)*(-x)^n/n ).
%C A255882 A210657(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial. In general it appears that when is k 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), A255883(k = 4) and A255884 (k = 6).
%H A255882 G. C. Greubel, <a href="/A255882/b255882.txt">Table of n, a(n) for n = 0..200</a>
%H A255882 E. W. Weisstein, <a href="https://mathworld.wolfram.com/EulerPolynomial.html">Euler Polynomial</a>
%F A255882 O.g.f.: exp( 2*x + 22*x^2/2 + 602*x^3/3 + 30742*x^4/4 + ... ) = 1 + 2*x + 13*x^2 + 224*x^3 + 8170*x^4 + ....
%F A255882 a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*3^(2*k)*E(2*k,1/3)*a(n-k).
%F A255882 a(n) ~ 2^(2*n + 2) * 3^(2*n + 1/2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - _Vaclav Kotesovec_, Jun 08 2019
%p A255882 #A255882
%p A255882 k := 3:
%p A255882 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 A255882 A210657[n_]:= 9^n EulerE[2 n, 1/3]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A210657[k]*(-x)^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 51}] (* _G. C. Greubel_, Aug 26 2018 *)
%Y A255882 Cf. A188514, A210657, A255881, A255883, A255884.
%K A255882 nonn,easy
%O A255882 0,2
%A A255882 _Peter Bala_, Mar 09 2015