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Search: id:A143922
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| A143922 |
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E.g.f. A(x) satisfies: A(x) = 1 + x*exp(Integral A(x)^2 dx). |
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+0
3
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| 1, 1, 2, 9, 52, 395, 3666, 40257, 510600, 7343523, 118093310, 2099660497, 40896662124, 866008634907, 19808285169834, 486698217317985, 12784410332144656, 357512156423101427, 10604399352362692182
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Compare definition of e.g.f. A(x) to the trivial statement:
if F(x) = 1/(1-x) then F(x) = 1 + x*exp(Integral F(x) dx).
Here Integral F(x) dx does not include the constant of integration.
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FORMULA
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E.g.f. derivative: A'(x) = [1 + x*A(x)^2]*(A(x) - 1)/x.
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EXAMPLE
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E.g.f. A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 395*x^5/5! +...
A(x)^2 = 1 + 2*x + 6*x^2/2! + 30*x^3/3! + 200*x^4/4! + 1670*x^5/5! +...
Let L(x) = Integral A(x)^2 dx where A(x) = 1 + x*exp(L(x)), then
L(x) = x + 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 200*x^5/5! +...
exp(L(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 611*x^5/5! +...
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PROGRAM
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*exp(intformal(A^2))); n!*polcoeff(A, n)}
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CROSSREFS
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Cf. A143923, A143924.
Sequence in context: A006152 A143508 A052882 this_sequence A110322 A161631 A121678
Adjacent sequences: A143919 A143920 A143921 this_sequence A143923 A143924 A143925
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Sep 06 2008
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