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A302700 E.g.f.: Sum_{n>=0} (exp(n*x) + 1)^n / (2 + exp(n*x))^(n+1).

%I A302700 #22 Aug 11 2018 11:06:04
%S A302700 1,1,13,385,21325,1898401,247841293,44611568065,10589093387725,
%T A302700 3204648461107681,1204384753185644173,550313048077989740545,
%U A302700 300436578515074737333325,193139598305033634851120161,144410707207961955130172624653,124258444226932649355925701301825,121911793079671988588136925596434125,135284324089583933279712302959420767841
%N A302700 E.g.f.: Sum_{n>=0} (exp(n*x) + 1)^n / (2 + exp(n*x))^(n+1).
%H A302700 Paul D. Hanna, <a href="/A302700/b302700.txt">Table of n, a(n) for n = 0..200</a>
%F A302700 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! equals:
%F A302700 (1) Sum_{n>=0} (exp(n*x) + 1)^n / (2 + exp(n*x))^(n+1).
%F A302700 (2) Sum_{n>=0} (exp(n*x) - 1)^n / (2 - exp(n*x))^(n+1).
%F A302700 (3) Sum_{n>=0} 2^n*exp(n^2*x/2)*cosh(n*x/2)^n/(1 + 2*exp(n*x/2)*cosh(n*x/2))^(n+1).
%F A302700 (4) Sum_{n>=0} 2^n*exp(n^2*x/2)*sinh(n*x/2)^n/(1 - 2*exp(n*x/2)*sinh(n*x/2))^(n+1).
%F A302700 a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.31165774853025500197969363638844... - _Vaclav Kotesovec_, Aug 10 2018
%e A302700 E.g.f.: A(x) = 1 + x + 13*x^2/2! + 385*x^3/3! + 21325*x^4/4! + 1898401*x^5/5! + 247841293*x^6/6! + 44611568065*x^7/7! + 10589093387725*x^8/8! + 3204648461107681*x^9/9! + ...
%e A302700 such that
%e A302700 A(x) = 1/3 + (exp(x)+1)/(2+exp(x))^2 + (exp(2*x)+1)^2/(2+exp(2*x))^3 + (exp(3*x)+1)^3/(2+exp(3*x))^4 + (exp(4*x)+1)^4/(2+exp(4*x))^5 + (exp(5*x)+1)^5/(2+exp(5*x))^6 + (exp(6*x)+1)^6/(2+exp(6*x))^7 + ...
%e A302700 Also,
%e A302700 A(x) = 1 + (exp(x)-1)/(2-exp(x))^2 + (exp(2*x)-1)^2/(2-exp(2*x))^3 + (exp(3*x)-1)^3/(2-exp(3*x))^4 + (exp(4*x)-1)^4/(2-exp(4*x))^5 + (exp(5*x)-1)^5/(2-exp(5*x))^6 + (exp(6*x)-1)^6/(2-exp(6*x))^7 + ...
%t A302700 nmax = 20; CoefficientList[Series[Sum[(E^(k*x) - 1)^k / (2 - E^(k*x))^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Aug 11 2018 *)
%o A302700 (PARI) {a(n) = my(A=1); A = sum(m=0,n+1, (exp(m*x + x*O(x^n)) - 1)^m / (2 - exp(m*x + x*O(x^n)))^(m+1) ); n!*polcoeff(A,n)}
%o A302700 for(n=0,30,print1(a(n),", "))
%Y A302700 Cf. A302598.
%K A302700 nonn
%O A302700 0,3
%A A302700 _Paul D. Hanna_, Apr 14 2018