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A052179
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Triangle of numbers arising in enumeration of walks on cubic lattice.
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33
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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; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 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
Riordan array ((1-4x-sqrt(1-8x+12x^2))/(2x^2),(1-4x-sqrt(1-8x+12x^2))/(2x)). Inverse of A159764. [From Paul Barry (pbarry(AT)wit.ie), 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
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LINKS
| R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
| 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
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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2009]
As an infinite lower triangular matrix = the Binomial transform of A091965 and 4-th bt of A053121. - Gary W. Adamson, Aug 03 2011
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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
[ From DELEHAM Philippe, Nov 04 2011]
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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}]] (* From Jean-François Alcover, Oct 10 2011, after Philippe Deleham *)
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CROSSREFS
| Cf. A053121, A064189, A039598, A091965
Sequence in context: A072651 A093035 A126791 * A171589 A126331 A013631
Adjacent sequences: A052176 A052177 A052178 * A052180 A052181 A052182
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KEYWORD
| nonn,walk,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 26 2000
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