0,2

Number of walks of length 2n on the 4-regular tree beginning and ending at some fixed vertex. The generating function for the corresponding sequence for the m-regular tree is 2*(m-1)/(m-2+m*sqrt(1-4*(m-1)*x)). When m=2 this reduces to the usual generating function for the central binomial coefficients. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Nov 11 2003

Main diagonal of the array A(0,j)=A(i,0)=1 for i,j>=0 and for i,j>=1 A(i,j)=min{A(i,j-1)+3*A(i-1,j); 3*A(i,j-1)+A(i-1,j)} - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 05 2004

Hankel transform is A133461 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 01 2007

a(n) = Sum{k, 0<=k<=n}A039599(n,k)*3^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 15 2009: (Start)

G.f.: 1/(1-4x*c(3x)), c(x) the g.f. of A000108;

G.f.: 1/(1-4x/(1-3x/(1-3x/(1-3x/(1-3x/(1-.... (continued fraction);

G.f.: 1/(1-4x-12x^2/(1-6x-9x^2/(1-6x-9x^2/(1-6x-9x^2/(1-... (continued fraction).

Integral representation: a(n)=(2/pi)*Int(x^n*sqrt(x(12-x))/(16-x),x,0,12). (End)

a(2)=28 because there are 4*4=16 walks whose second step is to return to the starting vertex and 4*3=12 walks whose second step is to move away from the starting vertex.

CoefficientList[ Series[3/(1 + 2Sqrt[1 - 12x]), {x, 0, 19}], x] (from Robert G. Wilson v Nov 12 2003)

Cf. A089022.

Adjacent sequences: A035607 A035608 A035609 this_sequence A035611 A035612 A035613

Sequence in context: A121203 A152599 A089023 this_sequence A046904 A030444 A093877

nonn

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