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A144301
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a(0) = a(1) = 1; thereafter a(n) = (2*n-3)*a(n-1) + a(n-2).
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5
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1, 1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 36496226977, 841146804577, 21065166341402, 569600638022431, 16539483668991901, 513293594376771362, 16955228098102446847
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| A variant of A001515, which is the main entry.
a(n) = number of increasing ordered trees on the vertex set [0,n] (counted by the double factorials A001147) in which n is the label on the leaf that terminates the leftmost path from the root. - David Callan, Aug 24 2011
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REFERENCES
| E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
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FORMULA
| a(n) = A001515(n-1) for n>= 1.
E.g.f.: A(x) = exp(1-sqrt(1-2*x)) satisfies A'(x) = A(x)/(1-sqrt(1-2*x)).
Hence a(n+1) = sum(k=0..n, a(n-k)*binomial(n,k)*(2*k)!/(k!*2^k) ).
A''(x) = (A'(x)/(1-2*x))*(1 + 1/sqrt(1-2*x)).
A''(x) = 2*x*A''(x) + A'(x) + A(x), which is equivalent to the recurrence in the definition.
a(n) = sum(k=0..n-1, binomial(n-1-k,2*k)*(2*k)!/(k!*2^k) ). [See Grosswald, p. 6, Eq. (8).]
a(n) ~ exp(1)*(2n-1)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]
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CROSSREFS
| See A001515 for much more about this sequence.
See A144498 for first differences.
Sequence in context: A125515 A135920 A001515 * A083659 A036247 A107877
Adjacent sequences: A144298 A144299 A144300 * A144302 A144303 A144304
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KEYWORD
| nonn,easy
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AUTHOR
| David Applegate and N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2008
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