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A162659 E.g.f. satisfies: A(x) = exp(x*A(x*A(x))). 2
1, 1, 3, 22, 281, 5396, 142297, 4865806, 207407489, 10710044776, 655655874641, 46789973764634, 3840103504940881, 358443042637767868, 37700333788138306937, 4432826052558222878206, 578707468284010393533953 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..16.

FORMULA

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! with a(0,m)=1, then

a(n,m) = Sum_{k=0..n} C(n,k) * m*(n-k+m)^(k-1) * a(n-k,k).

...

Let log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n!, then

L(n) = Sum_{k=1..n} C(n,k) * (n-k)^(k-1) * a(n-k,k).

...

E.g.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * [D^(n-1) A(x)^n] where operator D F(x) = d/dx x*F(x). - Paul D. Hanna, Mar 05 2013

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 281*x^4/4! + 5396*x^5/5! +...

A(x*A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 777*x^4/4! + 17581*x^5/5! +...

log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 196*x^4/4! + 3885*x^5/5! + 105486*x^6/6! +...

PROG

(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, binomial(n, k)*m*(n-k+m)^(k-1)*a(n-k, k))))}

(PARI) /* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */

{L(n)=if(n<1, 0, sum(k=1, n, binomial(n, k)*(n-k)^(k-1)*a(n-k, k)))}

CROSSREFS

Sequence in context: A074706 A293989 A141360 * A206801 A135862 A122778

Adjacent sequences:  A162656 A162657 A162658 * A162660 A162661 A162662

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 09 2009

STATUS

approved

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Last modified June 24 20:41 EDT 2021. Contains 345425 sequences. (Running on oeis4.)