A195513 - OEIS

%I #11 Jan 13 2014 13:05:32

%S 1,1,5,55,969,23471,722893,27024719,1188673041,60146184799,

%T 3441880575621,219796618046831,15495685980169849,1195427487905706095,

%U 100171535560697901693,9060375019110233758111,879801446255000326901025,91290178606709678636278463

%N E.g.f. satisfies: A(x) = exp(x*A(x)) - exp(x*A(x)^2) + exp(x*A(x)^3).

%H Vaclav Kotesovec, <a href="/A195513/b195513.txt">Table of n, a(n) for n = 0..100</a>

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

%F _ A(x) = Sum_{n>=0} x^n*(A(x)^n - A(x)^(2*n) + A(x)^(3*n)) / n!.

%F _ A(x) = C(x*A(x)) where C(x) = A(x/C(x)) equals the e.g.f. of A195512 and satisfies: C(x) = exp(x) - exp(x*C(x)) + exp(x*C(x)^2).

%F a(n) ~ n^(n-1) * sqrt(s*(exp(r*s) - exp(r*s^2)*s + exp(r*s^3)*s^2) / (exp(r*s)*r - 2*exp(r*s^2)*(1 + 2*r*s^2) + 3*exp(r*s^3)*s*(2 + 3*r*s^3))) / (exp(n) * r^n), where r = 0.149801756007819089... and s = 1.41946728153839215... are the roots of the equations r*(exp(r*s) + 3*exp(r*s^3)*s^2) = 1 + 2*exp(r*s^2)*r*s, and exp(r*s) + exp(r*s^3) = exp(r*s^2) + s. - _Vaclav Kotesovec_, Jan 13 2014

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23471*x^5/5! +...

%e Related series begin:

%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4 + 6476*x^5/5! + +...

%e exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 777*x^4 + 17521*x^5/5! +...

%e exp(x*A(x)^3) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1441*x^4 + 34516*x^5/5! +...

%e The e.g.f. of A195512 begins:

%e C(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4426*x^5/5! +...

%e where A(x/C(x)) = C(x) = exp(x) - exp(x*C(x)) + exp(x*C(x)^2).

%o (PARI) {a(n)=local(X=x+x*O(x^n));for(i=1,n,A=exp(X*A)-exp(X*A^2)+exp(X*A^3));n!*polcoeff(A,n)}

%o (PARI) {a(n)=local(X=x+x*O(x^n),A=1+X);for(i=1,n,A=sum(m=0,n,x^m*(A^m-A^(2*m)+A^(3*m))/m!));n!*polcoeff(A,n)}

%Y Cf. A195512, A195514.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 20 2011