A202668 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x) - (-1)^n)^n * x^n/n ).

1, 2, 4, 12, 42, 158, 618, 2498, 10360, 43832, 188420, 820608, 3613212, 16057640, 71933768, 324482500, 1472604586, 6719100254, 30804229858, 141829955338, 655541387406, 3040527731790, 14147444737654, 66018910398574, 308898542610666, 1448867831911170

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Formula

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 42*x^4 + 158*x^5 + 618*x^6 +...
where
log(A(x)) = (A(x) + 1)*x + (A(x) - 1)^2*x^2/2 + (A(x) + 1)^3*x^3/3 + (A(x) - 1)^4*x^4/4 +...
log(A(x)*(1-x*A(x))) = 1/(1 + x*A(x))*x + 1/(1 - x*A(x))^2*x^2/2 + 1/(1 + x*A(x))^3*x^3/3 + 1/(1 - x*A(x))^4*x^4/4 +...

Prog

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

Crossrefs

Cf. A202669, A185385, A202630, A202518, A155200.

Sequence in context: A126942 A126947 A188494 * A200222 A063179 A096802

Adjacent sequences: A202665 A202666 A202667 * A202669 A202670 A202671

Keyword

nonn

Author

Paul D. Hanna, Dec 22 2011

Status

approved