This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A064189 Triangle T(n,k), 0<=k<=n, read by rows, defined by: T(0,0)=1, T(n,k)= 0 if n
 1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 21, 30, 25, 14, 5, 1, 51, 76, 69, 44, 20, 6, 1, 127, 196, 189, 133, 70, 27, 7, 1, 323, 512, 518, 392, 230, 104, 35, 8, 1, 835, 1353, 1422, 1140, 726, 369, 147, 44, 9, 1, 2188, 3610, 3915, 3288, 2235, 1242, 560, 200, 54, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Motzkin triangle read in reverse order. T(n,k) = number of lattice paths from (0,0) to (n,k), staying weakly above the x-axis and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). Example: T(3,1) = 5 because we have HHU, UDU, HUH, UHH and UUD. Columns 0,1,2 and 3 give A001006 (Motzkin numbers), A002026 (first differences of Motzkin numbers), A005322 and A005323, respectively. - Emeric Deutsch, Feb 29 2004 Riordan array ((1-x-sqrt(1-2x-3x^2))/(2x^2), (1-x-sqrt(1-2x-3x^2))/(2x)). Inverse is the array (1/(1+x+x^2), x/(1+x+x^2)) (A104562). - Paul Barry, Mar 15 2005 Inverse binomial matrix applied to A039598 . - Philippe Deléham, Feb 28 2007 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)=T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-1,k+1) for k>=1 . - Philippe Deléham, Mar 27 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 Equals binomial transform of triangle A053121. [Gary W. Adamson, Oct 25 2008] Consider a semi-infinite chessboard with squares labeled (n,k), ranks or rows n >= 0, files or columns k >= 0; the number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,k). The recurrence relation given above relates to the movements of the king. This is essentially the comment made by Harrie Grondijs for the Motzkin triangle A026300. - Johannes W. Meijer, Oct 10 2010 REFERENCES See A026300 for additional references and other information. E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279. Sheng-Liang Yang et al., The Pascal rhombus and Riordan array, Fib. Q., 56:4 (2018), 337-347. See Fig. 3. LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015. I. Dolinka, J. East, R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015. Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019. Tom Halverson, Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018. Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3. Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091. FORMULA Sum_{k=0..n} T(n, k)*(k+1) = 3^n. Sum_{k=0..n} T(n, k)*T(n, n-k) = T(2*n, n) -T(2*n, n+2) G.f.: M/(1-t*z*M), where M=1+z*M+z^2*M^2 is the g.f. of the Motzkin numbers (A001006). - Emeric Deutsch, Feb 29 2004 Sum_{k>=0} T(m, k)*T(n, k) = A001006(m+n) . - Philippe Deléham, Mar 05 2004 Sum_{k>=0} T(n-k, k) = A005043(n+2) . - Philippe Deléham, May 31 2005 Column k has e.g.f. exp(x)*(BesselI(k,2*x)-BesselI(k+2,2*x)). - Paul Barry, Feb 16 2006 T(n,k) = sum{j=0..n, C(n,j)*(C(n-j,j+k)-C(n-j,j+k+2))}. - Paul Barry, Feb 16 2006 n-th row is generated from M^n * V, where M = the infinite tridiagonal matrix with all 1's in the super, main and subdiagonals; and V = the infinite vector [1,0,0,0...]. E.g. Row 3 = (4, 5, 3, 1), since M^3 * V = [4, 5, 3, 1, 0, 0, 0...]. - Gary W. Adamson, Nov 04 2006 T(n,k) = A122896(n+1,k+1) . - Philippe Deléham, Apr 21 2007 T(n,k) = k/n*sum(j=0..n, binomial(n,j)*binomial(j,2*j-n-k)). [Vladimir Kruchinin, Feb 12 2011] Sum_{k=0..n} T(n,k)*(-1)^k*(k+1) = (-1)^n. - Werner Schulte, Jul 08 2015 Sum_{k=0..n} T(n,k)*(k+1)^3 = (2*n+1)*3^n. - Werner Schulte, Jul 08 2015 G.f.: 2 / (1 - x + sqrt(1 - 2*x - 3*x^2) - 2*x*y) = Sum_{n >= k >=0} T(n, k) * x^n * y^k. - Michael Somos, Jun 06 2016 EXAMPLE Triangle begins: 1; 1,1; 2,2,1; 4,5,3,1; 9,12,9,4,1; ... Production matrix begins : 1, 1 1, 1, 1 0, 1, 1, 1 0, 0, 1, 1, 1 0, 0, 0, 1, 1, 1 0, 0, 0, 0, 1, 1, 1 0, 0, 0, 0, 0, 1, 1, 1 [Philippe Deléham, Nov 04 2011] MATHEMATICA T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 1, 1], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *) PROG (Sage) def A064189_triangel(dim):     M = matrix(SR, dim, dim)     for n in range(dim): M[n, n] = 1     for n in (1..dim-1):         for k in (0..n-1):             M[n, k] = M[n-1, k-1]+M[n-1, k]+M[n-1, k+1]     return M A064189_triangel(9) # Peter Luschny, Sep 20 2012 (PARI) {T(n, k) = if( k<0 || k>n, 0, polcoeff( polcoeff( 2 / (1 - x + sqrt(1 - 2*x - 3*x^2) - 2*x*y) + x * O(x^n), n), k))}; /* Michael Somos, Jun 06 2016 */ CROSSREFS Triangle in A026300 (the main entry for this sequence) with rows read in reverse order. Cf. A001006, A002026, A005322, A005323. Cf. A053121. - Gary W. Adamson, Oct 25 2008 Sequence in context: A183191 A273713 A322329 * A273897 A063415 A098977 Adjacent sequences:  A064186 A064187 A064188 * A064190 A064191 A064192 KEYWORD nonn,easy,tabl AUTHOR N. J. A. Sloane, Sep 21 2001 EXTENSIONS More terms from Vladeta Jovovic, Sep 23 2001 STATUS approved

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

Last modified October 23 17:32 EDT 2019. Contains 328373 sequences. (Running on oeis4.)