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A162659
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E.g.f. satisfies: A(x) = exp(x*A(x*A(x))).
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2
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1, 1, 3, 22, 281, 5396, 142297, 4865806, 207407489, 10710044776, 655655874641, 46789973764634, 3840103504940881, 358443042637767868, 37700333788138306937, 4432826052558222878206, 578707468284010393533953
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OFFSET
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0,3
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LINKS
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FORMULA
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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).
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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).
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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
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EXAMPLE
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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! +...
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PROG
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(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)))}
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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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