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Search: id:A161631
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| A161631 |
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E.g.f. satisfies: A(x) = 1 + x*exp(x*A(x)). |
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+0
2
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| 1, 1, 2, 9, 52, 425, 4206, 50827, 713000, 11500785, 208833850, 4226139731, 94226705772, 2296472176297, 60727113115046, 1732020500240955, 52998549321251536, 1731977581804704737, 60205422811336194546
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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E.g.f.: A(x) = 1 - LambertW(-x^2*exp(x))/x.
E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = 1 + x*exp(x)/B(x) = (1+sqrt(1+4*x*exp(x)))/2.
a(n) = n*A125500(n-1) for n>0, where exp(x*A(x)) = e.g.f. of A125500.
a(n) = n!*Sum_{k=0..n} C(n-k+1,k)/(n-k+1) * k^(n-k)/(n-k)!.
If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = n!*Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * k^(n-k)/(n-k)!.
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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! + 425*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +...
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PROGRAM
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(PARI) {a(n, m=1)=n!*sum(k=0, n, m*binomial(n-k+m, k)/(n-k+m)*k^(n-k)/(n-k)!)}
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CROSSREFS
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Cf. A125500.
Sequence in context: A052882 A143922 A110322 this_sequence A121678 A124347 A080146
Adjacent sequences: A161628 A161629 A161630 this_sequence A161632 A161633 A161634
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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), Jun 18 2009
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