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A108524
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Number of ordered rooted trees with n generators.
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1
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1, 2, 7, 32, 166, 926, 5419, 32816, 203902, 1292612, 8327254, 54358280, 358769152, 2390130038, 16051344307, 108548774240, 738563388214, 5052324028508, 34727816264050, 239733805643552, 1661351898336676, 11553558997057772
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A generator is a leaf or a node with just one child.
The Hankel transform of this sequence is 3^C(n+1,2). The Hankel transform of this sequence with 1 prepended (1,1,2,7,...) is 3^C(n,2). [From Paul Barry (pbarry(AT)wit.ie) 26 Jan 2011]
a(n) is the number of Schroder paths of semilength n-1 in which the (2,0)-steps that are not on the horizontal axis come in 2 colors. Example: a(3)=7 because we have HH, UDUD, UUDD, HUD, UDH, UBD, and URD, where U=(1,1), H=(2,0), D=(1,-1), while B and R are, respectively, blue and red (2,0)-steps. - Emeric Deutsch, May 02 2011
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LINKS
| Index entries for sequences related to rooted trees
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FORMULA
| G.f.: (sqrt(4*x^2-8*x+1) - 1)/(2*x-4)
G.f.: 1/(1-x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 10 2009]
a(n)=sum(i=1..n+1, (i*binomial(n+1,i)*sum(j=0..n-i, (-1)^j*2^(n-j)*binomial(n,j)*binomial(2*n-j-i-1,n-1)))/2^i)/(n*(n+1)); [Vladimir Kruchinin kru(AT)ie.tusur.ru, May 10 2011]
a(n) = upper left term in the following infinite square production matrix:
1, 1, 0, 0, 0,...
1, 1, 1, 0, 0,...
3, 3, 1, 1, 0,...
9, 9, 3, 1, 1,...
...
(where columns are (1, 1, 3, 9, 27, 81,...) prefaced with (0,0,1,2,3,...) zeros.
- Gary W. Adamson, Jul 11 2011
Conjecture: 2*n*a(n) +(24-17*n)*a(n-1) +4*(4*n-9)*a(n-2) +4*(3-n)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
G.f.: A(x)=(sqrt(4*x^2-8*x+1) - 1)/x/(2*x-4)= 1/(G(0)-x); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
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PROG
| (Maxima)
a(n):=sum((i*binomial(n+1, i)*sum((-1)^j*2^(n-j)*binomial(n, j)*binomial(2*n-j-i-1, n-1), j, 0, n-i))/2^i, i, 1, n+1)/(n*(n+1)); [Vladimir Kruchinin kru(AT)ie.tusur.ru, May 10 2011]
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CROSSREFS
| Cf. A108521-A108529, A108525, A000108, A001003.
Sequence in context: A168494 A181376 A183951 * A006781 A115197 A107593
Adjacent sequences: A108521 A108522 A108523 * A108525 A108526 A108527
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Jun 07 2005
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