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A268214 Number of paths from (0,0) to (n,n) that use E(1,0) and N(0,1) as steps and have odd number of East steps below the line y=x-1. 1
0, 0, 1, 4, 19, 72, 302, 1144, 4643, 17680, 70874, 271320, 1081742, 4160240, 16543932, 63871920, 253681955, 982571040, 3900295586, 15147970200, 60117323162, 234025162800, 928766304388, 3622709520144, 14378870714254, 56182491750112, 223031678941252 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..26.

Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

FORMULA

a(n) = binomial(2*n,n) - A268213(n).

G.f.: (8*x^2*g(x)-(f(x)-1)*(g(x)+1)+2*x*(-2+f(x)-3*g(x)+2*f(x)*g(x))) / (16*x^2*(4*x-1)), where f(x) = sqrt(1-4*x) and g(x) = sqrt(1+4*x).

PROG

(PARI) x='x+O('x^50); f=sqrt(1-4*x); g=sqrt(1+4*x); concat([0, 0], Vec((8*x^2*g-(f-1)*(g+1)-2*x*(2-f+3*g-2*f*g))/(16*x^4*(4*x-1)))) \\ Charles R Greathouse IV, Feb 01 2016

CROSSREFS

Cf. A000108, A268213.

Sequence in context: A267192 A219972 A284224 * A027208 A085348 A027232

Adjacent sequences:  A268211 A268212 A268213 * A268215 A268216 A268217

KEYWORD

nonn

AUTHOR

Ran Pan, Jan 28 2016

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.