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A078535
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Coefficients of power series that satisfies A(x)^6 - 36x*A(x)^7 = 1, A(0)=1.
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4
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1, 6, 162, 5760, 232254, 10077696, 458960580, 21634449408, 1046465787510, 51644846702592, 2590092194793948, 131621703842267136, 6762649550214036780
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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)=6^(2n)*binomial(7n/6-5/6, n)/(n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2002
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EXAMPLE
| A(x)^6 - 36x*A(x)^7 = 1 since A(x)^6 = 1 +36x +1512x^2 +68040x^3 +3193344x^4 +... and A(x)^7 = 1 +42x +1890x^2 +88704x^3 +... also a(5)=6^9, a(11)=6^22 = 131621703842267136.
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CROSSREFS
| Cf. A078531, A078532, A078533, A078534.
Sequence in context: A193370 A015086 A052466 * A177781 A178435 A183254
Adjacent sequences: A078532 A078533 A078534 * A078536 A078537 A078538
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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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