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A108447 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no peaks of the form ud. 10
1, 1, 4, 20, 113, 688, 4404, 29219, 199140, 1385904, 9807820, 70364704, 510609620, 3741212535, 27639233548, 205660399220, 1539916433473, 11594310041792, 87725707127600, 666681174728724, 5086601816592432, 38948589882247968 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Column 0 of A108446.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

FORMULA

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

G.f.: G satisfies G = 1 + z*G*(G^2+G-1).

a(n) = hypergeom([1-n,(n+3)/2,(n+4)/2],[2,n+3],-4) for n>=1. - Peter Luschny, Oct 30 2015

a(n) ~ sqrt((s-1) / (Pi*(1 + 3*s))) / (2*n^(3/2) * r^(n + 1/2)), where r = 0.1215851068721183026145063923222031450327682505108... and s = 1.451605962955776643742608112028547116887657025022... are real roots of the system of equations 1 + r*s*(-1 + s + s^2) = s, r*(-1 + 2*s + 3*s^2) = 1. - Vaclav Kotesovec, Nov 27 2017

O.g.f.: A(x) = (1/x) * Revert( x/c(x/(1 - x) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Mar 08 2020

EXAMPLE

a(2)=4 because we have uUddd, UddUdd, UdUddd and UUdddd.

MAPLE

a:=n->(1/n)*sum(binomial(n, j)*binomial(n+2*j, j-1), j=0..n): 1, seq(a(n), n=1..25);

a := n -> `if`(n=0, 1, simplify(hypergeom([1-n, (n+3)/2, (n+4)/2], [2, n+3], -4))): seq(a(n), n=0..21); # Peter Luschny, Oct 30 2015

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n, j]*Binomial[n + 2*j, j-1], {j, 0, n}]/n, {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 27 2017 *)

terms = 22; g[_] = 1; Do[g[x_] = 1+x*g[x]*(g[x]^2+g[x]-1) + O[x]^terms // Normal, {terms}]; CoefficientList[g[x], x] (* Jean-Fran├žois Alcover, Jul 19 2018 *)

CROSSREFS

Cf. A000108, A027307, A108446, A108425, A108426.

Sequence in context: A209200 A294119 A245375 * A287512 A211248 A028475

Adjacent sequences:  A108444 A108445 A108446 * A108448 A108449 A108450

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 10 2005

STATUS

approved

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Last modified December 4 02:18 EST 2020. Contains 338921 sequences. (Running on oeis4.)