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A064189
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Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if n < k, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1).
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56
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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
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OFFSET
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0,4
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COMMENTS
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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
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 from 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
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
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REFERENCES
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See A026300 for additional references and other information.
E. Barcucci, R. Pinzani, and R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
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LINKS
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R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
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FORMULA
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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
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) = (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
T(n,k) = binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4). - Peter Luschny, May 19 2021
The coefficients of the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + x + x^2)^n expanded about the point x = 0 give the entries in row n in reverse order. - Peter Bala, Sep 06 2022
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EXAMPLE
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Triangle begins:
[0] 1;
[1] 1, 1;
[2] 2, 2, 1;
[3] 4, 5, 3, 1;
[4] 9, 12, 9, 4, 1;
[5] 21, 30, 25, 14, 5, 1;
[6] 51, 76, 69, 44, 20, 6, 1;
[7] 127, 196, 189, 133, 70, 27, 7, 1;
[8] 323, 512, 518, 392, 230, 104, 35, 8, 1;
[9] 835, 1353, 1422, 1140, 726, 369, 147, 44, 9, 1.
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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 (End)
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MAPLE
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alias(C=binomial): A064189 := (n, k) -> add(C(n, j)*(C(n-j, j+k)-C(n-j, j+k+2)), j=0..n): seq(seq(A064189(n, k), k=0..n), n=0..10); # Peter Luschny, Dec 31 2019
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> simplify(hypergeom([1 -n/2, -n/2+1/2], [2], 4))); # Peter Luschny, Oct 08 2022
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MATHEMATICA
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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 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 2, 4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, May 19 2021 *)
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PROG
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(Sage)
M = matrix(ZZ, 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
(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 */
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CROSSREFS
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A026300 (the main entry for this sequence) with rows reversed.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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