login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A027307 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1). 58
1, 2, 10, 66, 498, 4066, 34970, 312066, 2862562, 26824386, 255680170, 2471150402, 24161357010, 238552980386, 2375085745978, 23818652359682, 240382621607874, 2439561132029314, 24881261270812490, 254892699352950850 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equals row sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna, Mar 30 2005

a(n) counts ordered complete ternary trees with 2*n-1 leaves, where the internal vertices come in two colors and such that each vertex and its rightmost child have different colors. See [Drake, Example 1.6.9]. An example is given below. - Peter Bala, Sep 29 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.

Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.

B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.6.9), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

J. Winter, M. M. Bonsangue and J. J. M. M. Rutten, Context-free coalgebras, 2013.

Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

FORMULA

G.f.: (2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3.

a(n) = (1/n) * Sum_{i=0..n-1} 2^(i+1)*binomial(2*n, i)*binomial(n, i+1)), n>0.

a(n) = 2*A034015(n-1), n>0.

a(n) = Sum_{k=0..n} C(2*n+k, n+2*k)*C(n+2*k, k)/(n+k+1). - Paul D. Hanna, Mar 30 2005

Given g.f. A(x), y=A(x)x satisfies 0=f(x, y) where f(x, y)=x(x-y)+(x+y)y^2 . - Michael Somos, May 23 2005

Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A085403(k)x^k).

G.f. A(x) satisfies A(x)=A006318(x*A(x)). - Vladimir Kruchinin, Apr 18 2011

The function B(x) = x*A(x^2) satisfies B(x) = x+x*B(x)^2+B(x)^3 and hence B(x) = compositional inverse of x*(1-x^2)/(1+x^2) = x+2*x^3+10*x^5+66*x^7+.... Let f(x) = (1+x^2)^2/(1-4*x^2+x^4) and let D be the operator f(x)*d/dx. Then a(n) equals 1/(2*n+1)!*D^(2*n)(f(x)) evaluated at x = 0. For a refinement of this sequence see A196201. - Peter Bala, Sep 29 2011

D-finite with recurrence: 2*n*(2*n+1)*a(n) = (46*n^2-49*n+12)*a(n-1) - 3*(6*n^2-26*n+27)*a(n-2) - (n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ sqrt(50+30*sqrt(5))*((11+5*sqrt(5))/2)^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012

EXAMPLE

a(3) = 10. Internal vertices colored either b(lack) or w(hite); 5 uncolored leaf vertices shown as o.

........b...........b.............w...........w.....

......./|\........./|\.........../|\........./|\....

....../.|.\......./.|.\........./.|.\......./.|.\...

.....b..o..o.....o..b..o.......w..o..o.....o..w..o..

..../|\............/|\......../|\............/|\....

.../.|.\........../.|.\....../.|.\........../.|.\...

..o..o..o........o..o..o....o..o..o........o..o..o..

....................................................

........b...........b.............w...........w.....

......./|\........./|\.........../|\........./|\....

....../.|.\......./.|.\........./.|.\......./.|.\...

.....w..o..o.....o..w..o.......b..o..o.....o..b..o..

..../|\............/|\......../|\............/|\....

.../.|.\........../.|.\....../.|.\........../.|.\...

..o..o..o........o..o..o....o..o..o........o..o..o..

....................................................

........b...........w..........

......./|\........./|\.........

....../.|.\......./.|.\........

.....o..o..w.....o..o..b.......

........../|\........./|\......

........./.|.\......./.|.\.....

........o..o..o.....o..o..o....

...............................

MATHEMATICA

a[n_] := ((n+1)*(2n)!*Hypergeometric2F1[-n, 2n+1, n+2, -1]) / (n+1)!^2; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 14 2011, after Pari *)

PROG

(PARI) a(n)=if(n<1, n==0, sum(i=0, n-1, 2^(i+1)*binomial(2*n, i)*binomial(n, i+1))/n)

(PARI) a(n)=sum(k=0, n, binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1)) \\ Paul D. Hanna

(PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1) ) /* Michael Somos, May 23 2005 */

CROSSREFS

Cf. A104978. A196201.

Sequence in context: A230050 A278459 A278461 * A278460 A278462 A060206

Adjacent sequences:  A027304 A027305 A027306 * A027308 A027309 A027310

KEYWORD

nonn

AUTHOR

Emeric Deutsch

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 15 08:53 EDT 2020. Contains 336487 sequences. (Running on oeis4.)