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A047891
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Number of planar rooted trees with n nodes and tricolored end nodes.
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15
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1, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322, 15361896, 100389306, 663180024, 4421490924, 29712558576, 201046204173, 1368578002188, 9366084668802, 64403308499592, 444739795023054, 3082969991029800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also number of lattice paths from (0,0) to (n,n), with steps (1,0),(0,1) and (1,1), that never rise above the line y=x and the steps (1,1) are colored red or blue. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 28 2003
The Hankel transform (see A001906 for definition) of this sequence forms A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ... ] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2006
With a(0)=0, this is the series reversion of x(1-x)/(1+2x). [From Paul Barry (pbarry(AT)wit.ie), Oct 18 2009]
Row sums of the Riordan matrix A121576. [Emanuele Munarini, May 18 2011]
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REFERENCES
| Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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LINKS
| Index entries for sequences related to rooted trees
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FORMULA
| (1-2z-sqrt(4z^2-8z+1))/2z.
For n>0, a(n)=(1/n)*sum(k=0, n, 3^k*C(n, k)*C(n, k-1)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003
a(1)=1, a(n)=2*a(n-1)+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004
The Hankel transform (see A001906 for definition) of this sequence form A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ... ] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2006
2*a(n)=A054872(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2007
Contribution from Paul Barry (pbarry(AT)wit.ie), Feb 01 2009: (Start)
G.f.: 1/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);
a(n)=sum{k=0..n, C(n+k,2k)*2^(n-k)*A000108(k)}. (End)
G.f.: 1/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Oct 18 2009]
a(0) = 1, for n>=1, a(n) = 3*A007564(n) [From Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009]
From Emanuele Munarini, May 18 2011: (Start)
a(n) = sum(binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k,k=0..n)/2.
Recurrence: (n+3)*(n+4)*a(n+3)-6*(n+3)^2*a(n+2)-12*(n+1)^2*a(n+1)+8*n*(n+1)*a(n)=0. (End)
G.f.: A(x)=(1-2*x-sqrt(4*x^2-8*x+1))/(2*x)= (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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MAPLE
| A047891_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A047891_list(20); #Peter Luschny, May 19 2011
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MATHEMATICA
| CoefficientList[Series[(1-2x-Sqrt[1-8x+4x^2])/(2x), {x, 0, 100}], x]
[Emanuele Munarini, May 18 2011]
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PROG
| (PARI) a(n)=if(n<1, 1, sum(k=0, n, 3^k*binomial(n, k)*binomial(n, k-1))/n)
(Maxima) makelist(sum(binomial(n, k)*binomial(2*n-k+1, n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k, k, 0, n)/2, n, 0, 24); [Emanuele Munarini, May 18 2011]
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CROSSREFS
| Essentially the same as A025231.
Cf. A006318, A121576.
Sequence in context: A133158 A194089 A178807 * A166991 A151498 A103370
Adjacent sequences: A047888 A047889 A047890 * A047892 A047893 A047894
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KEYWORD
| nonn,eigen,easy
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AUTHOR
| Louis Shapiro (lshapiro(AT)howard.edu)
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EXTENSIONS
| More terms from Christian Bower, Dec 11 1999
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