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 A047891 Number of planar rooted trees with n nodes and tricolored end nodes. 20
 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; text; internal format)
 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. P. Barry, A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations , J. Int. Seq. 14 (2011) # 11.3.8 Z. Chen, 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, 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. Eric Weisstein's MathWorld, Legendre Polynomial. 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], , 3) - 2*0^(n-1). (End) 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 := 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:=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: A328295 A194089 A178807 * A166991 A276366 A243521 Adjacent sequences:  A047888 A047889 A047890 * A047892 A047893 A047894 KEYWORD nonn,eigen,easy AUTHOR EXTENSIONS More terms from Christian G. Bower, Dec 11 1999 STATUS approved

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)