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A078532
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Coefficients of power series that satisfies A(x)^3 - 9x*A(x)^4 = 1, A(0)=1.
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4
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1, 3, 27, 315, 4158, 59049, 880308, 13586859, 215233605, 3479417370, 57168561996, 951892141473, 16026585711660, 272383068872700, 4666865660812044, 80521573261807755, 1397858693681272230, 24398716826612190447, 427921056863230599900, 7537621933880388620010
(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)=3^(2n)*binomial(4n/3-2/3, n)/(n+1) - EmericDeutsch(AT)msn.com (deutsch(AT)duke.poly.edu), Dec 10 2002
Sequence with offset 1 is expansion of reversion of g.f. x*(1-9*x)^(1/3), which equals x times the g.f. of A004990.
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EXAMPLE
| A(x)^3 - 9x*A(x)^4 = 1 since A(x)^3 = 1 +9x +108x^2 +1458x^3 +21060x^4 +... and A(x)^4 = 1 +12x +162x^2 +2340x^3 +... also a(2)=3^3, a(5)=3^10.
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MATHEMATICA
| Table[3^(2n) Binomial[(4n-2)/3, n]/(n+1), {n, 0, 20}] (* From Harvey P. Dale, Nov 03 2011 *)
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CROSSREFS
| Cf. A078531, A078533, A078534, A078535.
Cf. A004990.
Sequence in context: A127503 A204821 A200903 * A153853 A067000 A168593
Adjacent sequences: A078529 A078530 A078531 * A078533 A078534 A078535
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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, Nov 03 2011
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