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A015084
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q-Catalan numbers (recurrence version) for q=3.
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0
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1, 1, 4, 43, 1252, 104098, 25511272, 18649337311, 40823535032644, 267924955577741566, 5274102955963545775864, 311441054994969341088610030, 55171471477692117486494217498280
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Limit_{n->inf} a(n)/3^((n-1)(n-2)/2) = Product{k=1..inf} 1/(1-1/3^k) = 1.785312341998534190367486296013703535718796... - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2005
It appears that the Hankel transform is 3^A002412(n). [From Paul Barry (pbarry(AT)wit.ie), Aug 01 2008]
Hankel transform of the aerated sequence is 3^C(n+1,3). [From Paul Barry (pbarry(AT)wit.ie), Oct 31 2008]
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FORMULA
| a(n) = sum_{i=1}^{n-1} q^{(i-1)} a(i) a(n-i).
G.f. satisfies: A(x) = 3*x/(3-A(3*x)) = x/(1-x/(1-3*x/(1-3^2*x/(1-3^3*x/(1-...))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2005
a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
1, 3, 0, 0, 0, 0,...
1, 3, 9, 0, 0, 0,...
1, 3, 9, 27, 0, 0,...
1, 3, 9, 27, 81, 0,...
...
- Gary W. Adamson, Jul 14 2011
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PROG
| (PARI) a(n)=if(n==1, 1, sum(i=1, n-1, 3^(i-1)*a(i)*a(n-i))) (Hanna)
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CROSSREFS
| Sequence in context: A197717 A152282 A153255 * A176827 A102388 A071125
Adjacent sequences: A015081 A015082 A015083 * A015085 A015086 A015087
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KEYWORD
| nonn
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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EXTENSIONS
| More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2005
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