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A129400 Number of walks of length n on one 60-degree wedge of the equilateral triangular lattice. The walk can go along the walls of the wedge, but cannot cross the walls. 4
1, 2, 8, 32, 144, 672, 3264, 16256, 82688, 427520, 2240512, 11874304, 63533056, 342712320, 1861779456, 10176823296, 55932813312, 308907737088, 1713473323008, 9541666209792, 53322206674944, 298943898451968 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Counts colored Motzkin paths where each of the steps has two possible colors. Series reversion of x/(1+2x+4x^2). - Paul Barry, Sep 04 2007

Hankel transform is 4^C(n+1,2). - Paul Barry, Oct 01 2009

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

FORMULA

a(n) = 2^n*A001006(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*2^(n-2k)*2^k*2^k where C(n) = A000108(n). - Paul Barry, Sep 04 2007

G.f.: 1/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-.... (continued fraction). - Paul Barry, Oct 01 2009

G.f.: (1/(8*x^2)) * (1-2*x-(1-4*x-12*x^2)^(1/2)) - Mark van Hoeij, Nov 02 2009

E.g.f.: a(n) = n! * [x^n] exp(2*x)*BesselI(1,4*x)/(2*x). - Peter Luschny, Aug 25 2012

Recurrence: (n+2)*a(n) = 2*(2*n+1)*a(n-1) + 12*(n-1)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 3*sqrt(3)*6^n/(2*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012

a(n) = 2^n*GegenbauerC(n, -n-1, -1/2)/(n+1). - Peter Luschny, May 09 2016

EXAMPLE

a(1) = 2 because we can go east or northeast.

MAPLE

countwalk2 := proc (i::integer, j::integer, n::integer) option remember: if n < 0 or j < 0 or i < j then 0 elif n = 0 and i = 0 and j = 0 then 1 elif n = 0 then 0 else procname(i-2, j, n-1)+procname(i+2, j, n-1)+procname(i-1, j+1, n-1)+procname(i+1, j+1, n-1)+procname(i+1, j-1, n-1)+procname(i-1, j-1, n-1) end if end proc: counter2 := proc (n::nonnegint) option remember: add(add(countwalk2(i, j, n), i = 0 .. 2*n), j = 0 .. n) end proc:

g := n -> simplify(2^n*GegenbauerC(n, -n-1, -1/2)/(n+1)):

seq(g(n), n=0..21); # Peter Luschny, May 09 2016

T := proc(n, k) option remember;

if n < 0 or k < 0 then 0

elif n = 0 then binomial(2*k, k)/(k+1)

else 2*(T(n-1, k+1) - T(n-1, k)) fi end:

a := n -> T(n, 1): seq(a(n), n=0..21); # Peter Luschny, Aug 23 2017

MATHEMATICA

CoefficientList[Series[1/(8*x^2)*(1-2*x-Sqrt[1-4*x-12*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

CROSSREFS

Sequence in context: A150862 A150863 A084137 * A003304 A150864 A202814

Adjacent sequences:  A129397 A129398 A129399 * A129401 A129402 A129403

KEYWORD

nonn,walk

AUTHOR

Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 28 2007

STATUS

approved

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Last modified August 18 01:04 EDT 2019. Contains 326059 sequences. (Running on oeis4.)