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A216136
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E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^3).
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7
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1, 1, 6, 75, 1456, 38560, 1296786, 52935680, 2542934024, 140557741488, 8787984793440, 613224873661752, 47245653830341176, 3983499665690137944, 364844394810538703256, 36070922050704987248280, 3828821598701561543783616, 434302348322255060713797120
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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(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
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EXAMPLE
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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! +...
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n, m=1)=sum(k=0, n, m*(3*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*(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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