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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

%I

%S 0,0,1,4,19,72,302,1144,4643,17680,70874,271320,1081742,4160240,

%T 16543932,63871920,253681955,982571040,3900295586,15147970200,

%U 60117323162,234025162800,928766304388,3622709520144,14378870714254,56182491750112,223031678941252

%N 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.

%H Ran Pan, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.

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

%F 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).

%o (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

%Y Cf. A000108, A268213.

%K nonn

%O 0,4

%A _Ran Pan_, Jan 28 2016

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Last modified November 11 22:31 EST 2019. Contains 329046 sequences. (Running on oeis4.)