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A209881
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G.f. satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x))].
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5
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1, 1, 4, 21, 136, 1030, 8856, 84861, 894928, 10291986, 128165720, 1718395602, 24686953968, 378444958060, 6167922926704, 106525443913245, 1943838547593888, 37375737467294362, 755393226726677976, 16011417246585359046, 355187993770520180400, 8230524179585799932820
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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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Given g.f. A(x), the g.f. of A075834 = 1 + x/(1 - x*A(x)).
Forms the logarithmic derivative of A075834.
O.g.f. A(x) satisfies: [x^n] ( 1 + x/(1 - x*A(x)) )^(n+1) = (n+1)! for n>=0.
O.g.f. A(x) satisfies: [x^n] exp( n * Integral A(x) dx ) * (n + 1 - A(x)) = 0 for n > 0. - Paul D. Hanna, Jun 04 2018
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 136*x^4 + 1030*x^5 + 8856*x^6 +...
The g.f. of A075834, G(x) = 1/(1 - x*A(x)), begins:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 206*x^5 + 1476*x^6 +...
The logarithm of the g.f. of A075834 begins:
log(G(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, 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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