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A143154
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E.g.f.: A(x) = x + log(1 - A(x))^2.
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1
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1, 2, 18, 262, 5320, 138728, 4419156, 166319424, 7221397848, 355312006392, 19537581248592, 1187337791554176, 79025863405440432, 5716937001401316000, 446654003380859659488, 37480492611898380241248
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Radius of convergence is r = (-1 + 6*A(r) - A(r)^2)/4 = 0.172815973872...
where A(r) = 1 - exp((A(r)-1)/2) = 0.2965325775...
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FORMULA
| E.g.f.: A(x) = Series_Reversion( x - log(1 - x)^2 ).
E.g.f. derivative: A'(x) = (1 - A(x))/(1 - A(x) + 2*log(1 - A(x))).
a(n)=((n-1)!*sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, (-1)^(n+j-1)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(2*(j-l))!*stirling1(n-l+j-1,2*(j-l)))/(n-l+j-1)!)))), n>0. [From Vladimir Kruchinin, Feb 07 2012]
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EXAMPLE
| A(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...
-log(1 - A(x)) = G(x) = the g.f. of A143155:
G(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! +...
G(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...
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PROG
| (PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + log(1-A)^2); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(serreverse(x-log(1-x+x*O(x^n))^2), n)}
(Maxima) a(n):=((n-1)!*sum(binomial(n+k-1, n-1)*sum((-1)^(n+j-1)*binomial(k, j)*sum((binomial(j, l)*(2*(j-l))!*stirling1(n-l+j-1, 2*(j-l)))/(n-l+j-1)!, l, 0, j), j, 0, k), k, 0, n-1)); [From Vladimir Kruchinin, Feb 07 2012]
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CROSSREFS
| Cf. A143155, A143138.
Sequence in context: A151362 A099880 A141009 * A032037 A138275 A127134
Adjacent sequences: A143151 A143152 A143153 * A143155 A143156 A143157
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KEYWORD
| nonn,changed
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jul 27 2008
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