A259267 E.g.f. A(x) satisfies: A'(x) = exp(2*A(A(x))).

1, 2, 12, 128, 2016, 42656, 1145280, 37563008, 1464675840, 66533778944, 3466031815680, 204489094565888, 13524452573872128, 994257291909816320, 80668058806271016960, 7179145234347383128064, 697131195162680465817600, 73522035747248454761578496, 8387016414085244676889116672

Offset: 1

Author

Paul D. Hanna, Jul 03 2015

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 2016*x^5/5! +...
Related expansions:
A'(x) =  1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2016*x^4/4! + 42656*x^5/5! +...
A(A(x)) = log(A'(x))/2 = x + 4*x^2/2! + 36*x^3/3! + 520*x^4/4! + 10512*x^5/5! + 276064*x^6/6! + 8987712*x^7/7! + 351278080*x^8/8! +...
The exponential of e.g.f. A(x) equals the e.g.f. of A233336:
exp(A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 3097*x^5/5! + 63963*x^6/6! + 1677883*x^7/7! +...+ A233336(n)*x^n/n! +...
		

Crossrefs

Cf. A233336, A214645.

Programs

PARI

{a(n)=local(A=x+x^2); for(i=0, n, A=intformal(exp(2*subst(A, x, A+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

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

(1) A''(x) = 2*exp( 4*A(A(x)) + 2*A(A(A(x))) ).

(2) exp(-2*A(x)) = d/dx Series_Reversion(A(x)).

(3) A(x) = log(F(x)) where F(x) satisfies: F( Integral 1/F(x)^2 dx ) = exp(x) and equals the e.g.f. of A233336.

a(n) = 2^(n-1) * A214645(n) for n>=1.