login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A052179 Triangle of numbers arising in enumeration of walks on cubic lattice. 33
1, 4, 1, 17, 8, 1, 76, 50, 12, 1, 354, 288, 99, 16, 1, 1704, 1605, 700, 164, 20, 1, 8421, 8824, 4569, 1376, 245, 24, 1, 42508, 48286, 28476, 10318, 2380, 342, 28, 1, 218318, 264128, 172508, 72128, 20180, 3776, 455, 32, 1, 1137400, 1447338 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Triangle T(n,k), 0<=k<=n, read by rows given by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0) = 4*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k+1) for k>=1. - Philippe Deléham, Mar 27 2007

Triangle read by rows:T(n,k)=number of lattice paths from (0,0) to (n,k)that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and four types of steps H=(1,0); example: T(3,1)=50 because we have UDU, UUD, 16 HHU paths, 16 HUH paths and 16 UHH paths. - Philippe Deléham, Sep 25 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k>=1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

Riordan array ((1-4x-sqrt(1-8x+12x^2))/(2x^2),(1-4x-sqrt(1-8x+12x^2))/(2x)). Inverse of A159764. - Paul Barry, Apr 21 2009

6^n = (n-th row terms) dot (first n+1 terms in (1,2,3,...)). Example: 6^3 =  216 = (76, 50, 12, 1) dot (1, 2, 3, 4) = (76 + 100 + 36 + 4) = 216. - Gary W. Adamson, Jun 15 2011

A subset of the "family of triangles" (Deleham comment of Sep 25 2007) is the succession of Binomial transforms beginning with triangle A053121, (0,0); giving -> A064189, (1,1); -> A039598, (2,2); -> A091965, (3,3); -> A052179, (4,4); -> A125906, (5,5) ->, etc; generally the binomial transform of the triangle generated from (n,n) = that generated from ((n+1),(n+1)). - Gary W. Adamson, Aug 03 2011

LINKS

Table of n, a(n) for n=0..46.

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

FORMULA

Sum_{k, k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A005572(m+n). - Philippe Deléham, Sep 15 2005

n-th row = M^n * V, where M = the infinite tridiagonal matrix with all 1's in the super and subdiagonals and (4,4,4...) in the main diagonal. E.g., Row 3 = (76, 50, 12, 1) since M^3 * V = [76, 50, 12, 1, 0, 0, 0...]. - Gary W. Adamson, Nov 04 2006

Sum_{k, 0<=k<=n}T(n,k)=A005573(n). - Philippe Deléham, Feb 04 2007

Sum_{k, 0<=k<=n}T(n,k)*(k+1)=6^n. - Philippe Deléham, Mar 27 2007

Sum_{k, 0<=k<=n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x= -2, -1, 0, 1 respectively. - Philippe Deléham, Nov 28 2009

As an infinite lower triangular matrix = the Binomial transform of A091965 and 4th Binomial transform of A053121. - Gary W. Adamson, Aug 03 2011

EXAMPLE

Triangle begins:

1;

4,1;

17,8,1;

76,50,12,1;

354,288,99,16,1; ...

Production matrix begins:

4, 1

1, 4, 1

0, 1, 4, 1

0, 0, 1, 4, 1

0, 0, 0, 1, 4, 1

0, 0, 0, 0, 1, 4, 1

0, 0, 0, 0, 0, 1, 4, 1

- Philippe Deléham, Nov 04 2011

MATHEMATICA

t[0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0; t[n_, 0] := t[n, 0] = 4*t[n-1, 0] + t[n-1, 1]; t[n_, k_] := t[n, k] = t[n-1, k-1] + 4*t[n-1, k] + t[n-1, k+1]; Flatten[ Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Oct 10 2011, after _Philippe Deleham_ *)

CROSSREFS

Cf. A053121, A064189, A039598, A091965.

Sequence in context: A209411 A093035 A126791 * A171589 A126331 A013631

Adjacent sequences:  A052176 A052177 A052178 * A052180 A052181 A052182

KEYWORD

nonn,walk,tabl,easy,nice

AUTHOR

N. J. A. Sloane, Jan 26 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 19 04:00 EDT 2015. Contains 256803 sequences.