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A191412
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G.f. satisfies: A(x) = x*exp( Sum_{n>=1} A(x^n/(1-x^n))/n ).
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1
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1, 1, 3, 9, 34, 139, 643, 3198, 17186, 98438, 598551, 3842893, 25957607, 183858842, 1361853786, 10523285935, 84648613474, 707461189513, 6132612218266, 55050390426042, 510994289786018, 4898133459331104, 48424584171850411
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OFFSET
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1,3
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COMMENTS
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Compare the g.f. to the g.f. P(x) of the partition numbers (A000041):
* P(x) = exp( Sum_{n>=1} (x^n/(1-x^n))/n ),
and to the g.f. R(x) of rooted trees with n nodes (A000081):
* R(x) = x*exp( Sum_{n>=1} R(x^n)/n ).
Consider also the trivial identity:
* B(x) = x*exp( Sum_{n>=1} B(x^n/(1+x^n))/n ) where B(x) = x/(1-x).
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LINKS
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 34*x^5 + 139*x^6 + 643*x^7 +...
where the g.f. A(x) satisfies:
log(A(x)/x) = A(x/(1-x)) + A(x^2/(1-x^2))/2 + A(x^3/(1-x^3))/3 + A(x^4/(1-x^4))/4 +...
Explicitly, the logarithmic series begins:
log(A(x)/x) = x + 5*x^2/2 + 19*x^3/3 + 93*x^4/4 + 466*x^5/5 + 2633*x^6/6 + 15534*x^7/7 + 97645*x^8/8 +...
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PROG
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(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x*exp(sum(m=1, n, subst(A, x, x^m/(1-x^m+x*O(x^n)))/m))); polcoeff(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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