A195511 E.g.f. satisfies: A(x) = (exp(x) + exp(x*A(x)^2))/2.
1, 1, 3, 19, 197, 2801, 50407, 1098371, 28122761, 827684785, 27534518411, 1021777860995, 41847737874637, 1875044409274817, 91239372967844207, 4791502346638758931, 270114113377777911569, 16269795487513345957025, 1042794341136010753491475
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 197*x^4/4! + 2801*x^5/5! +... Related series begin: exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 37*x^3/3! + 393*x^4/4! + 5601*x^5/5! +... A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 600*x^4/4! + 8712*x^5/5! +... The g.f. C(x) of A195510 begins: C(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 156*x^5/5! + 1561*x^6/6! +... where A(x/C(x)) = C(x) = (exp(x*C(x)) + exp(x/C(x)))/2.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..100
Programs
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PARI
{a(n)=local(X=x+x*O(x^n),A=1+X);for(i=1,n,A=(exp(X)+exp(X*A^2))/2);n!*polcoeff(A,n)}
Formula
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * (1 + A(x)^(2*n))/2.
E.g.f. satisfies: A(x) = C(x*A(x)) where C(x) = A(x/C(x)) is the g.f. of A195510 and satisfies: C(x) = (exp(x*C(x)) + exp(x/C(x)))/2.
a(n) ~ sqrt(((1+2*r)*s^2-1)/(2+4*r*s^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.258248183317928786953777... and s = 1.7522591181936492232545... are the roots of the equations exp(r) + exp(r*s^2) = 2*s, exp(r*s^2)*r*s = 1. - Vaclav Kotesovec, Jan 11 2014