A195513 E.g.f. satisfies: A(x) = exp(x*A(x)) - exp(x*A(x)^2) + exp(x*A(x)^3).

1, 1, 5, 55, 969, 23471, 722893, 27024719, 1188673041, 60146184799, 3441880575621, 219796618046831, 15495685980169849, 1195427487905706095, 100171535560697901693, 9060375019110233758111, 879801446255000326901025, 91290178606709678636278463

Offset

0,3

Links

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

Formula

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

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

_ 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).

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

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23471*x^5/5! +...
Related series begin:
exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4 + 6476*x^5/5! + +...
exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 777*x^4 + 17521*x^5/5! +...
exp(x*A(x)^3) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1441*x^4 + 34516*x^5/5! +...
The e.g.f. of A195512 begins:
C(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4426*x^5/5! +...
where A(x/C(x)) = C(x) = exp(x) - exp(x*C(x)) + exp(x*C(x)^2).

Prog

(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)}
(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)}

Crossrefs

Cf. A195512, A195514.

Sequence in context: A357394 A093352 A293013 * A172493 A155807 A135861

Adjacent sequences: A195510 A195511 A195512 * A195514 A195515 A195516

Keyword

nonn

Author

Paul D. Hanna, Sep 20 2011

Status

approved