A289739 Expansion of solution to dy/dx = y + exp(y).

0, 1, 2, 5, 17, 79, 474, 3468, 29799, 293528, 3258373, 40234231, 546921835, 8115147998, 130503876054, 2260929219675, 41979302557200, 831593152814251, 17506400133530765, 390278100156698627, 9185223726173708408, 227578002295869672508, 5921091852493279814589

Offset: 0

Author

Michael Somos, Aug 09 2017

Keywords

• nonn

Comments

See A266539 for more details.

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 0..435

Crossrefs

Cf. A266329.

Programs

Maple

S:= dsolve({diff(y(x),x) = y(x) + exp(y(x)), y(0)=0},y(x),series,order=31):
seq(coeff(rhs(S),x,j)*j!,j=0..30); # Robert Israel, Aug 09 2017

Mathematica

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

PARI

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

PARI

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

Formula

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

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