A195512 E.g.f. satisfies: A(x) = exp(x) - exp(x*A(x)) + exp(x*A(x)^2).

1, 1, 3, 22, 269, 4426, 91567, 2289918, 67241113, 2268437842, 86469399731, 3675752021206, 172416288958597, 8846409344413434, 492872054023465495, 29633162309495166526, 1912378764997833270065, 131856366022646024614306, 9673570273675513393639387

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Formula

E.g.f. satisfies:

_ A(x) = Sum_{n>=0} x^n*(1 - A(x)^n + A(x)^(2*n)) / n!.

_ A(x) = B(x/A(x)) where B(x) = A(x*B(x)) equals the e.g.f. of A195513 and satisfies: B(x) = exp(x*B(x)) - exp(x*B(x)^2) + exp(x*B(x)^3).

a(n) ~ n^(n-1) * sqrt((-exp(r) + exp(r*s)*s - exp(r*s^2)*s^2) / (exp(r*s)*r - 2*exp(r*s^2)*(1 + 2*r*s^2))) / (exp(n) * r^n), where r = 0.2257106995256572853... and s = 1.621740007241874226... are the roots of the equations 1 + exp(r*s)*r = 2*exp(r*s^2)*r*s, and exp(r) + exp(r*s^2) = exp(r*s) + s. - Vaclav Kotesovec, Jan 13 2014

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4426*x^5/5! +...
Related series begin:
exp(x*A(x)) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2136*x^5/5! +...
exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! + 6561*x^5/5! +...
The e.g.f. of A195513 begins:
B(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23471*x^5/5! +...
where A(x*B(x)) = B(x) = exp(x*B(x)) - exp(x*B(x)^2) + exp(x*B(x)^3).

Prog

(PARI) {a(n)=local(X=x+x*O(x^n), A=1+X); for(i=1, n, A=exp(X)-exp(X*A)+exp(X*A^2)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(X=x+x*O(x^n), A=1+X); for(i=1, n, A=sum(m=0, n, x^m*(1-A^m+A^(2*m))/m!)); n!*polcoeff(A, n)}

Crossrefs

Cf. A195513, A195514.

Sequence in context: A242794 A367181 A005264 * A052892 A155806 A196022

Adjacent sequences: A195509 A195510 A195511 * A195513 A195514 A195515

Keyword

nonn

Author

Paul D. Hanna, Sep 20 2011

Status

approved