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Search: id:A052179
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| A052179 |
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Triangle of numbers arising in enumeration of walks on cubic lattice. |
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28
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| 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; listen)
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OFFSET
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0,2
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COMMENT
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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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007
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LINKS
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R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
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Sum_{k, k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A005572(m+n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 15 2005
Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A005572(m+n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 28 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 (qntmpkt(AT)yahoo.com), Nov 04 2006
Sum_{k, 0<=k<=n}T(n,k)=A005573(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 04 2007
Sum_{k, 0<=k<=n}T(n,k)*(k+1)=6^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
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EXAMPLE
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1; 4,1; 17,8,1; 76,50,12,1; 354,288,99,16,1; ...
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CROSSREFS
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Adjacent sequences: A052176 A052177 A052178 this_sequence A052180 A052181 A052182
Sequence in context: A072651 A093035 A126791 this_sequence A126331 A013631 A113355
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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njas, Jan 26 2000
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