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Search: id:A078534
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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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5
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| 1, 5, 100, 2625, 78125, 2502500, 84150000, 2929265625, 104646953125, 3814697265625, 141323284375000, 5305403695312500, 201382633183593750
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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
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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
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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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CROSSREFS
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Cf. A078531, A078532, A078533, A078535.
Sequence in context: A088995 A093749 A128784 this_sequence A141120 A123668 A090436
Adjacent sequences: A078531 A078532 A078533 this_sequence A078535 A078536 A078537
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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), Nov 28 2002
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