A245309 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * (d/dx x*A(x)^n)^n / A(x)^(n^2).

1, 1, 3, 25, 397, 10101, 372991, 18744853, 1227094905, 101320257097, 10294575759451, 1262050509059121, 183700770307306693, 31322680620408105085, 6184922808789945458967, 1400325997347499801032301, 360395936189117983848624241, 104632853179210298481432557073

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..130

Formula

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

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

(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..[n/2]} C(n-k,k) * (n-k)^k * A'(x)^k/A(x)^k / (n-k)!.

a(n) ~ c * (n!)^2, where c = 0.881770167... . - Vaclav Kotesovec, Jul 25 2014

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 397*x^4/4! + 10101*x^5/5! +...
where
A(x) = 1 + x*(d/dx x*A(x))/A(x) + x^2/2!*(d/dx x*A(x)^2)^2/A(x)^4 + x^3/3!*(d/dx x*A(x)^3)^3/A(x)^9 + x^4/4!*(d/dx x*A(x)^4)^4/A(x)^16 +...
or, equivalently,
A(x) = 1 + x*(1 + x*A'(x)/A(x)) + x^2*(1 + 2*x*A'(x)/A(x))^2/2! + x^3*(1 + 3*x*A'(x)/A(x))^3/3! + x^4*(1 + 4*x*A'(x)/A(x))^4/4! +...
Related series:
A'(x)/A(x) = 1 + 2*x + 18*x^2/2! + 300*x^3/3! + 7980*x^4/4! + 305520*x^5/5! + 15801240*x^6/6! + 1058302560*x^7/7! + 88992343440*x^8/8! +...

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*deriv(x*A^m)^m/A^(m^2)/m!+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*(1 + m*x*A'/A)^m/m!+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A304858 A192552 A143925 * A074708 A323217 A160143

Adjacent sequences: A245306 A245307 A245308 * A245310 A245311 A245312

Keyword

nonn

Author

Paul D. Hanna, Jul 23 2014

Status

approved