A216135 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^2).

1, 1, 4, 33, 424, 7440, 165846, 4487966, 142930376, 5237697744, 217106129040, 10043789510832, 513016686849624, 28676264198255856, 1741205465305623240, 114124985340571809480, 8030944551164700156096, 603905270121593669417472, 48328182913534662635924544

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, 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. - Vaclav Kotesovec, Dec 28 2013

Links

Seiichi Manyama, Table of n, a(n) for n = 0..357

Formula

(1) a(n) = Sum_{k=0..n} (2*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*(2*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*(2*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*(2*n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.

Limit n->infinity a(n)^(1/n)/n = exp(2*(1-r)/(r-2))*(2-r+exp(r/(2-r))) = 1.7802115440907..., where r = 0.655269699533064... is the root of the equation exp(r/(2-r)) = (r-2)/r*(r + LambertW(-1,-r*exp(-r)). - Vaclav Kotesovec, Sep 17 2013

a(n) ~ s*sqrt((s^s-1)/(2*(s^s-1)*(2*s^s-1)-s)) * n^(n-1) * (s^(2+s)/(s^s-1))^n / exp(n), where s = 1.627893875694537903318580987... is the root of the equation (1+log(s))*s = 2*(s^s-1). - Vaclav Kotesovec, Dec 28 2013

Example

E.g.f. A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 424*x^4/4! + 7440*x^5/5! +...
where
A(x)^A(x) = 1 + x + 6*x^2/2! + 60*x^3/3! + 864*x^4/4! + 16360*x^5/5! +...
1/(1-x*A(x)^2) = 1 + x + 6*x^2/2! + 60*x^3/3! + 864*x^4/4! + 16360*x^5/5! +...

Mathematica

Table[Sum[(2*n-k+1)^(k-1)*(-1)^(n-k)*StirlingS1[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 17 2013 *)

Prog

(PARI) a(n, m=1)=sum(k=0, n, m*(2*n-k+m)^(k-1)*(-1)^(n-k)*stirling(n, k, 1));
for(n=0, 21, print1(a(n), ", "))
(PARI) {a(n, m=1)=sum(k=0, n, m*(2*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^2+x*O(x^n)))/A)); n!*polcoeff(A, n)}
for(n=0, 21, print1(a(n), ", "))

Crossrefs

Cf. A141209, A216136, A229237.

Sequence in context: A360234 A111534 A162655 * A052885 A277184 A192548

Adjacent sequences: A216132 A216133 A216134 * A216136 A216137 A216138

Keyword

nonn

Author

Paul D. Hanna, Sep 01 2012

Status

approved