A185385 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2*A(x) - (-1)^n)^n * x^n/n ).

1, 3, 11, 61, 381, 2527, 17559, 126265, 931321, 7007035, 53568131, 414929621, 3249392917, 25684315319, 204645707183, 1641910625009, 13253684541553, 107561523423731, 877109999610107, 7183095973808493, 59053492869471661, 487189276030904207, 4032100262853037127

Offset: 0

Author

Paul D. Hanna, Dec 22 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 3*x + 11*x^2 + 61*x^3 + 381*x^4 + 2527*x^5 + 17559*x^6 +...
where
log(A(x)) = (2*A(x) + 1)*x + (2*A(x) - 1)^2*x^2/2 + (2*A(x) + 1)^3*x^3/3 + (2*A(x) - 1)^4*x^4/4 +...
log(A(x)*(1-2*x*A(x))) = 1/(1 + 2*x*A(x))*x + 1/(1 - 2*x*A(x))^2*x^2/2 + 1/(1 + 2*x*A(x))^3*x^3/3 + 1/(1 - 2*x*A(x))^4*x^4/4 +...
		

Crossrefs

Cf. A202519, A163138, A202668, A202630, A202518, A155200.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (2*A-(-1)^m+x*O(x^n))^m*x^m/m))); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = 1/(1-2*x*A(x)) * exp( Sum_{n>=1} 1/(1 - (-1)^n*2*x*A(x))^n * x^n/n ).

G.f. satisfies: A(x) = sqrt( (1 - (2*A(x)+1)^2*x^2)/(1 - (2*A(x)-1)^2*x^2) ) / (1 - (2*A(x)+1)*x).

G.f. satisfies: 0 = -(1+x) - 2*x*A(x) + (1+x)*(1-x)^2*A(x)^2 - 2*x*(1-x)^2*A(x)^3 - 2^2*x^2*(1+x)*A(x)^4 + 2^3*x^3*A(x)^5.