A289739 - OEIS

%I #18 Aug 21 2017 13:21:36

%S 0,1,2,5,17,79,474,3468,29799,293528,3258373,40234231,546921835,

%T 8115147998,130503876054,2260929219675,41979302557200,831593152814251,

%U 17506400133530765,390278100156698627,9185223726173708408,227578002295869672508,5921091852493279814589

%N Expansion of solution to dy/dx = y + exp(y).

%C See A266539 for more details.

%H Robert Israel, <a href="/A289739/b289739.txt">Table of n, a(n) for n = 0..435</a>

%F E.g.f. y(x) = log(A(x)) and y'(x) = B(x) where A(x), B(x) are as in A266539.

%F a(n) ~ c^n * (n-1)!, where c = 1/Integral_{x=0..infinity} 1/(x + exp(x)) dx = 1.2400861064984976662394901721056528110217273471501174317019052800276... - _Vaclav Kotesovec_, Aug 21 2017

%e E.g.f. = x + 2*x^2/2! + 5*x^3/3! + 17*x^4/4! + ...

%p S:= dsolve({diff(y(x),x) = y(x) + exp(y(x)), y(0)=0},y(x),series,order=31):

%p seq(coeff(rhs(S),x,j)*j!,j=0..30); # _Robert Israel_, Aug 09 2017

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ InverseSeries[ Series[Integrate[ 1 / (x + Exp[x]), x], {x, 0, n}]], {x, 0, n}]];

%o (PARI) {a(n) = if( n<0, 0, my(A = O(x)); for(k=1, n, A = intformal(A + exp(A))); n! * polcoeff(A, n))};

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( serreverse( intformal( 1 / (exp(x + x * O(x^n)) + x))), n))};

%Y Cf. A266329.

%K nonn

%O 0,3

%A _Michael Somos_, Aug 09 2017