0,2

Also number of paths from (0,0) to (n,0) in an n X n grid using only Northeast, East and Southeast steps and the East steps come in four colors. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 03 2002

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 153

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

N. J. A. Sloane, Transforms

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Generating function A(x) satisfies 1+(xA)^2=A-4xA.

a(0)=1 and, for n>0, a(n)=4a(n-1)+ Sum[a(i-1)a(n-i-1), i=1, n-1] - John W. Layman (layman(AT)math.vt.edu), Jan 07 2000.

G.f.: (1-4*x-sqrt(1-8*x+12*x^2))/(2*x^2).

a(n)=((2*n+1)*a(n-1)-3*(n-1)*a(n-2))*4/(n+2), n>0.

a(m+n) = Sum_{k, k>=0} A052179(m, k)*A052179(n, k) = A052179(m+n, 0). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 15 2005

a(n) = 4a(n-1)+A052177(n-1) = A052179(n, 0) = 6*A005573(n)-A005573(n-1) = sum{j = 0, ..., [n/2]}(4^(n-2j)*C(n, 2j)*C(2j, j)/(j+1))). - Henry Bottomley (se16(AT)btinternet.com), Aug 23 2001

(PARI) a(n)=polcoeff((1-4*x-sqrt(1-8*x+12*x^2+x^3*O(x^n)))/2, n+2)

Binomial transform of A002212. Cf. A001006.

Sequence shifted right twice is A025228.

Sequence in context: A081910 A026773 A081186 this_sequence A081922 A124325 A151248

Adjacent sequences: A005569 A005570 A005571 this_sequence A005573 A005574 A005575

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Michael Somos, Jun 10, 2000.