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A101785
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G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2).
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3
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1, 1, 1, 2, 5, 12, 30, 79, 213, 584, 1628, 4600, 13138, 37871, 110043, 321978, 947813, 2805104, 8341608, 24912004, 74686460, 224694128, 678143656, 2052640752, 6229616730, 18952875247, 57792705415, 176596786934, 540679385663
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Formula may be derived using the Lagrange Inversion theorem (cf. A049124).
a(n) = number of Dyck n-paths (A000108) all of whose descents have odd length. For example, a(3) counts UUUDDD, UDUDUD. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
The number of noncrossing partitions of [n] with all blocks of odd size. E.g.: a(4)=5 with the five partitions being 123/4, 124/3, 134/2,1/234 and 1/2/3/4. - Lou Shapiro (lshapiro(AT)howard.edu), Jan 07 2006
Number of ordered trees with n edges in which every non-leaf vertex has an odd number of children. - David Callan (callan(AT)stat.wisc.edu), Mar 30 2007
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FORMULA
| a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1, k)*C(n, 2*k)/(2*k+1) for n>0, with a(0)=1. G.f.: A(x) = [ series reversion of x*(1-x^2)/(1+x-x^2) ]/x.
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EXAMPLE
| Generated from Fibonacci polynomials (A011973) and
coefficients of odd powers of 1/(1-x):
a(1) = 1*1/1
a(2) = 1*1/1 + 0*1/3
a(3) = 1*1/1 + 1*3/3
a(4) = 1*1/1 + 2*6/3 + 0*1/5
a(5) = 1*1/1 + 3*10/3 + 1*5/5
a(6) = 1*1/1 + 4*15/3 + 3*15/5 + 0*1/7
a(7) = 1*1/1 + 5*21/3 + 6*35/5 + 1*7/7
a(8) = 1*1/1 + 6*28/3 + 10*70/5 + 4*28/7 + 0*1/9
This process is equivalent to the formula:
a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1,k)*C(n,2*k)/(2*k+1).
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PROG
| (PARI) {a(n)=if(n==0, 1, sum(k=0, (n-1)\2, binomial(n-k-1, k)*binomial(n, 2*k)/(2*k+1)))}
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CROSSREFS
| Cf. A011973, A049124.
Sequence in context: A103287 A136704 A120895 * A003089 A179544 A112412
Adjacent sequences: A101782 A101783 A101784 * A101786 A101787 A101788
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Dec 16 2004
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