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A078533
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Coefficients of power series that satisfies A(x)^4 - 16x*A(x)^5 = 1, A(0)=1.
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4
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1, 4, 56, 1024, 21216, 473088, 11075328, 268435456, 6677665280, 169514369024, 4373549027328, 114349209288704, 3023068543631360, 80675644291153920
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).
If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002
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FORMULA
| a(n)=4^(2n)*binomial(5n/4-3/4, n)/(n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2002
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EXAMPLE
| A(x)^4 - 16x*A(x)^5 = 1 since A(x)^4 = 1 +16x +320x^2 +7040x^3 +163840x^4 +... and A(x)^5 = 1 +20x +440x^2 +10240x^3 +... also a(3)=4^5, a(7)=4^14=268435456.
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CROSSREFS
| Cf. A078531, A078532, A078534, A078535.
Sequence in context: A191466 A130219 A111874 * A009058 A061924 A183612
Adjacent sequences: A078530 A078531 A078532 * A078534 A078535 A078536
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2002
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