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A078534
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Coefficients of power series that satisfies A(x)^5 - 25x*A(x)^6 = 1, A(0)=1.
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4
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1, 5, 100, 2625, 78125, 2502500, 84150000, 2929265625, 104646953125, 3814697265625, 141323284375000, 5305403695312500, 201382633183593750, 7715985752343750000, 298023223876953125000, 11591412585295166015625, 453601640704152832031250
(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)=5^(2n)*binomial(6n/5-4/5, n)/(n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2002
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EXAMPLE
| A(x)^5 - 25x*A(x)^6 = 1 since A(x)^5 = 1 +25x +750x^2 +24375x^3 +831250x^4 +... and A(x)^6 = 1 +30x +975x^2 +33250x^3 +... also a(4)=5^7, a(9)=5^18 = 3814697265625.
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MATHEMATICA
| Table[5^(2n) Binomial[(6n-4)/5, n]/(n+1), {n, 0, 25}] (* From Harvey P. Dale, Mar 27 2011 *)
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CROSSREFS
| Cf. A078531, A078532, A078533, A078535.
Sequence in context: A147539 A156276 A128784 * A141120 A123668 A197200
Adjacent sequences: A078531 A078532 A078533 * A078535 A078536 A078537
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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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EXTENSIONS
| More terms from Harvey P. Dale, Mar 27 2011.
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