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 A052179 Triangle of numbers arising in enumeration of walks on cubic lattice. 34
 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" (Deléham 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 G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2. R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6. FORMULA Sum_{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..n} T(n,k) = A005573(n). - Philippe Deléham, Feb 04 2007 Sum_{k=0..n} T(n,k)*(k+1) = 6^n. - Philippe Deléham, Mar 27 2007 Sum_{k=0..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 G.f.: 2/(1 - 4*x - 2*x*y + sqrt(1 - 8*x + 12*x^2)). - Daniel Checa, Aug 17 2022 G.f. for the m-th column: x^m*(A(x))^(m+1), where A(x) is the g.f. of the sequence counting the walks on the cubic lattice starting and finishing on the xy plane and never going below it (A005572). Explicitly, the g.f. is x^m*((1 - 4*x - sqrt(1 - 8*x + 12*x^2))/(2*x^2))^(m+1). - Daniel Checa, Aug 28 2022 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 MAPLE T:= proc(n, k) option remember; `if`(min(n, k)<0, 0, `if`(max(n, k)=0, 1, T(n-1, k-1)+4*T(n-1, k)+T(n-1, k+1))) end: seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Oct 28 2021 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. A039598, A053121, A064189, A091965. Sequence in context: A093035 A301624 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

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Last modified November 30 18:31 EST 2023. Contains 367461 sequences. (Running on oeis4.)