login
E.g.f. A(x) satisfies: A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.
0

%I #5 Oct 26 2016 17:33:21

%S 1,4,36,508,10020,253804,7853076,287078908,12106864260,578586544204,

%T 30901130685876,1823983173981148,117911755067635620,

%U 8284976875099852204,628692318063511556436,51240154266491883376828,4464155216699369664399300,414013560595951627772296204,40722939746084736801890208756

%N E.g.f. A(x) satisfies: A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.

%F a(n) = 2*A143138(n) for n>1.

%F E.g.f. A(x) satisfies:

%F (1) A(x) = x + 2 * (exp( (A(x) + x)/2 ) - 1)^2.

%F (2) A(x) = -x + 2 * Series_Reversion( x - (exp(x)-1)^2 ).

%F (3) A(x) = x + 2 * Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^(2*n) / n!.

%F (4) A( log(1+x) - x^2 ) = log(1+x) + x^2.

%e E.g.f.: A(x) = x + 4*x^2/2! + 36*x^3/3! + 508*x^4/4! + 10020*x^5/5! + 253804*x^6/6! + 7853076*x^7/7! + 287078908*x^8/8! + 12106864260*x^9/9! + 578586544204*x^10/10! +...

%e such that A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.

%o (PARI) {a(n) = n!*polcoeff( -x + 2*serreverse( x - (exp(x +x*O(x^n)) - 1)^2 ), n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A143138.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 15 2016