A047891 Number of planar rooted trees with n nodes and tricolored end nodes.

1, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322, 15361896, 100389306, 663180024, 4421490924, 29712558576, 201046204173, 1368578002188, 9366084668802, 64403308499592, 444739795023054, 3082969991029800

Offset

1,2

Comments

Essentially the same as A025231.

Also number of lattice paths from (0,0) to (n-1,n-1), 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, May 28 2003

The Hankel transform (see A001906 for definition) of this sequence forms A049656(n+1) = [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006

With a(0)=0, this is the series reversion of x(1-x)/(1+2x). - Paul Barry, Oct 18 2009

Row sums of the Riordan matrix A121576. - Emanuele Munarini, May 18 2011

References

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.

Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO] (2016), eq. (1.13), a=3, b=1.

Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Luis Verde-Star A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Eric Weisstein's MathWorld, Legendre Polynomial.

Index entries for sequences related to rooted trees

Formula

G.f.: (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/2.

For n>0, a(n+1) = (1/n)*Sum_{k=0..n} 3^k*C(n, k)*C(n, k-1) - Benoit Cloitre, May 10 2003

a(1)=1, a(n) = 2*a(n-1) + Sum_{i=1..(n-1)} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004

The Hankel transform (see A001906 for definition) of this sequence form A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006

2*a(n) = A054872(n+1). - Philippe Deléham, Aug 17 2007

From Paul Barry, Feb 01 2009: (Start)

G.f.: x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);

a(n+1) = Sum_{k=0..n} C(n+k,2k)*2^(n-k)*A000108(k). (End)

G.f.: x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). - Paul Barry, Oct 18 2009

a(1) = 1, for n>=1, a(n+1) = 3*A007564(n). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009

From Emanuele Munarini, May 18 2011: (Start)

a(n+1) = (Sum_{k=0..n} 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)/2.

D-finite with recurrence: (n+2)*(n+3)*a(n+3) - 6*(n+2)^2*a(n+2) - 12*(n)^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 = 1 - G(0); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012

G.f.: x/W(0), where W(k)= k+1 - 2*x*(k+1) - x*(k+1)*(k+2)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013

From Vladimir Reshetnikov, Nov 01 2015: (Start)

a(n) = 2^(n-1)*(LegendreP_n(2) - LegendreP_{n-2}(2))/(2n-1).

a(n) = 3*hypergeom([1-n,2-n], [2], 3) - 2*0^(n-1). (End)

a(n) = 2^(n-1)*hypergeom([1-n, n], [2], -1/2). - Peter Luschny, Nov 25 2020

a(n) ~ 3^(1/4) * (1 + sqrt(3))^(2*n - 1) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 31 2021

D-finite with recurrence n*a(n) +4*(-2*n+3)*a(n-1) +4*(n-3)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

Example

G.f. = x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1686*x^6 + 9912*x^7 + ...

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

Mathematica

CoefficientList[Series[(1-2x-Sqrt[1-8x+4x^2])/(2x), {x, 0, 100}], x] (* Emanuele Munarini, May 18 2011 *)
a[ n_] := SeriesCoefficient[(1 - 2 x - Sqrt[1 - 8 x + 4 x^2]) / 2, {x, 0, n}]; (* Michael Somos, Apr 10 2014 *)
Table[2^(n-1) (LegendreP[n, 2] - LegendreP[n-2, 2])/(2n-1), {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
Table[3 Hypergeometric2F1[1-n, 2-n, 2, 3] - 2 KroneckerDelta[n-1], {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

Prog

(PARI) a(n)=if(n<2, n==1, n--; sum(k=0, n, 3^k*binomial(n, k)*binomial(n, k-1))/n)
(PARI) x='x+O('x^100); Vec((1-2*x-sqrt(1-8*x+4*x^2))/2) \\ Altug Alkan, Nov 02 2015
(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 */
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-2*x-Sqrt(1-8*x+4*x^2))/(2*x))) // G. C. Greubel, Feb 10 2018

Crossrefs

Cf. A006318, A121576, A054872.

Sequence in context: A194089 A178807 A361844 * A166991 A276366 A243521

Adjacent sequences: A047888 A047889 A047890 * A047892 A047893 A047894

Keyword

nonn,eigen,easy

Author

Louis Shapiro

Extensions

More terms from Christian G. Bower, Dec 11 1999

Status

approved