OFFSET
0,2
COMMENTS
Exponential convolution of Catalan numbers and powers of 2. - Vladeta Jovovic, Dec 03 2004
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 4 colors. Example: a(3)=37 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 4 paths of shape UHD. - Emeric Deutsch, May 02 2011
a(n) is the number of Schroeder paths of semilength n in which the (2,0)-steps come in 2 colors and having no (2,0)-steps at levels 1,3,5,... - José Luis Ramírez Ramírez, Mar 30 2013
From Tom Copeland, Nov 08 2014: (Start)
This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t)=x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A126930.)
O.g.f.: G(x) = C[P[P(x,-1),-1]] = C[P(x,-2)] = [1-sqrt(1-4*x/(1-2x)]/2 = x*A064613(x).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv:1110.6638 [math.NT], 2011.
FORMULA
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)*2^(n-k)/(k+1).
a(n) = 2^n*hypergeom([1/2, -n], [2], -2).
G.f.: (1-sqrt((1-6*x)/(1-2*x)))/(2*x). - Vladeta Jovovic, May 03 2003
With offset 1: a(1) = 1, a(n) = 2^(n-1) + Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
D-finite with recurrence (n+1)*a(n) = (8*n-2)*a(n-1) - (12*n-12)*a(n-2). - Vladeta Jovovic, Jul 16 2004
E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x)). - Vladeta Jovovic, Dec 03 2004
Inverse binomial transform of A104455. - Philippe Deléham, Nov 30 2007
G.f.: 1/(1-3*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-... (continued fraction). - Paul Barry, Jul 02 2009
a(n) = Sum_{0<=k<=n} A052179(n,k)*(-1)^k. - Philippe Deléham, Nov 28 2009
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
3, 1, 0, 0, ...
1, 3, 1, 0, ...
1, 1, 3, 1, ...
1, 1, 1, 3, ...
... (End)
a(n) ~ 2^(n-3/2)*3^(n+3/2)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013
G.f. A(x) satisfies: A(x) = 1/(1 - 2*x) + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020
MATHEMATICA
CoefficientList[Series[(1-Sqrt[(1-6*x)/(1-2*x)])/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 29 2013 *)
a[n_] := 2^n Hypergeometric2F1[1/2, -n, 2, -2];
Array[a, 22, 0] (* Peter Luschny, Jan 27 2020 *)
PROG
(PARI) x='x+O('x^66); Vec((1-sqrt((1-6*x)/(1-2*x)))/(2*x)) /* Joerg Arndt, Mar 31 2013 */
(Magma) I:=[3, 10]; [1] cat [n le 2 select I[n] else ((8*n-2)*Self(n-1)-(12*n-12)*Self(n-2))div (n+1): n in [1..30]]; // Vincenzo Librandi, Jan 23 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Sep 24 2001
EXTENSIONS
Name clarified using a comment of Vladeta Jovovic by Peter Bala, Jan 27 2020
STATUS
approved