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A216136 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^3). 3
1, 1, 6, 75, 1456, 38560, 1296786, 52935680, 2542934024, 140557741488, 8787984793440, 613224873661752, 47245653830341176, 3983499665690137944, 364844394810538703256, 36070922050704987248280, 3828821598701561543783616, 434302348322255060713797120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

More generally, if G(x) = 1/(1 - x*G(x)^p)^(G(x)^q), then

[x^n/n! ] G(x)^m = Sum_{k=0..n} m*(p*n+q*k+m)^(k-1) * (-1)^(n-k)*Stirling1(n,k), and

[x^n/n! ] log(G(x)) = Sum_{k=1..n} (p*n+q*k)^(k-1) * (-1)^(n-k)*Stirling1(n,k).

Generally, for A(x)^A(x) = 1/(1-x*A(x)^p) is limit n->infinity a(n)^(1/n)/n = exp(p*(1-r)/(r-p))*(p-r+exp(r/(p-r))), where r is the root of the equation exp(r/(p-r)) = (r-p)/r*(r + LambertW(-1,-r*exp(-r)). - Vaclav Kotesovec, Sep 17 2013

Generally, if e.g.f. A(x) satisfies A(x)^A(x) = 1/(1-x*A(x)^p), then a(n) ~ s*sqrt((s^s-1)/(p*(s^s-1)*(p*s^s-1)-s)) * n^(n-1) * (s^(p+s)/(s^s-1))^n / exp(n), where s is the root of the equation (1+log(s))*s = (s^s-1)*p. Compared with my previous result, limit n->infinity a(n)^(1/n)/n = s^(p+s)/(s^s-1)/exp(1). - Vaclav Kotesovec, Dec 28 2013

LINKS

Table of n, a(n) for n=0..17.

FORMULA

(1) a(n) = Sum_{k=0..n} (3*n-k+1)^(k-1)* (-1)^(n-k)* Stirling1(n,k).

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

(2) a(n,m) = Sum_{k=0..n} m*(3*n-k+m)^(k-1) * (-1)^(n-k) *Stirling1(n,k) ;

which is equivalent to the following:

(3) a(n,m) = Sum_{k=0..n} m*(3*n-k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1+j*x)};

(4) a(n,m) = n!*Sum_{k=0..n} m*(3*n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.

Limit n->infinity a(n)^(1/n)/n = exp(3*(1-r)/(r-3))*(3-r+exp(r/(3-r))) = 2.685525290558..., where r = 0.77397865498224... is the root of the equation exp(r/(3-r)) = (r-3)/r*(r + LambertW(-1,-r*exp(-r)). - Vaclav Kotesovec, Sep 17 2013

a(n) ~ s*sqrt((s^s-1)/(3*(s^s-1)*(3*s^s-1)-s)) * n^(n-1) * (s^(3+s)/(s^s-1))^n / exp(n), where s = 1.4158017407588097722625060603... is the root of the equation (1+log(s))*s = 3*(s^s-1). - Vaclav Kotesovec, Dec 28 2013

EXAMPLE

E.g.f. A(x) = 1 + x + 6*x^2/2! + 75*x^3/3! + 1456*x^4/4! + 38560*x^5/5! +...

where

A(x)^A(x) = 1 + x + 8*x^2/2! + 114*x^3/3! + 2388*x^4/4! + 66480*x^5/5! +...

1/(1-x*A(x)^3) = 1 + x + 8*x^2/2! + 114*x^3/3! + 2388*x^4/4! + 66480*x^5/5! +...

MATHEMATICA

Table[Sum[(3*n-k+1)^(k-1)*(-1)^(n-k)*StirlingS1[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 17 2013 *)

PROG

(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

{a(n, m=1)=sum(k=0, n, m*(3*n-k+m)^(k-1)*(-1)^(n-k)*Stirling1(n, k) )}

for(n=0, 21, print1(a(n), ", "))

(PARI) {a(n, m=1)=sum(k=0, n, m*(3*n-k+m)^(k-1)*polcoeff(prod(j=1, n-1, 1+j*x), n-k))}

for(n=0, 21, print1(a(n), ", "))

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(-log(1-x*(A^3+x*O(x^n)))/A)); n!*polcoeff(A, n)}

for(n=0, 21, print1(a(n), ", "))

CROSSREFS

Cf. A141209, A216135, A229237.

Sequence in context: A139088 A193784 A162863 * A126462 A081066 A185289

Adjacent sequences:  A216133 A216134 A216135 * A216137 A216138 A216139

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 01 2012

STATUS

approved

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Last modified October 24 05:38 EDT 2021. Contains 348217 sequences. (Running on oeis4.)