A097609 Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k horizontal steps at level 0.

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 2, 3, 0, 1, 6, 7, 3, 4, 0, 1, 15, 14, 12, 4, 5, 0, 1, 36, 37, 24, 18, 5, 6, 0, 1, 91, 90, 67, 36, 25, 6, 7, 0, 1, 232, 233, 165, 106, 50, 33, 7, 8, 0, 1, 603, 602, 438, 264, 155, 66, 42, 8, 9, 0, 1, 1585, 1586, 1147, 719, 390, 215, 84, 52, 9, 10, 0, 1

Offset

0,8

Comments

Row sums give the Motzkin numbers (A001006).

Column 0 is A005043.

Riordan array ((1+x-sqrt(1-2*x-3*x^2))/(2*x*(1-x)), (1+x-sqrt(1-2*x-3*x^2))/(2*(1-x))). - Paul Barry, Jun 21 2008

Inverse of Riordan array ((1-x)/(1-x+x^2), x*(1-x)/(1-x+x^2)), which is A104597. - Paul Barry, Jun 21 2008

Triangle read by rows, product of A064189 and A130595 considered as infinite lower triangular arrays; A097609 = A064189*A130195 = B*A053121*B^(-1) where B = A007318. - Philippe Deléham, Dec 07 2009

T(n+1,1) = A187306(n). - Philippe Deléham, Jan 28 2014

The number of lattice paths from (0,0) to (n,k) that do not cross below the x-axis and use up-step=(1,1) and down-steps=(1,-z) where z is a positive integer. For example, T(4,0) = 3: [(1,1)(1,1)(1,-1)(1,-1)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,1)(1,-3)]. - Nicholas Ham, Aug 20 2015

The convolution triangle of the Riordan numbers A005043. - Peter Luschny, Oct 09 2022

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).

Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

Igor Dolinka, James East, and Robert D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Trends in Mathematics 2000, pp 127-139.

Formula

G.f.: 2/(1 -2*t*z +z +sqrt(1-2*z-3*z^2)).

T(n,k) = T(n-1,k-1)+ Sum_{j>=1} T(n-1,k+j) with T(0,0)=1. - Philippe Deléham, Jan 23 2010

T(n,k) = (k/n)*Sum_{j=k..n} (-1)^(n-j)*C(n,j)*C(2*j-k-1,j-1), n>0. - Vladimir Kruchinin, Feb 05 2011

Example

Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  1, 2, 0, 1;
  3, 2, 3, 0, 1;
  6, 7, 3, 4, 0, 1;
Row n has n+1 terms.
T(5,2) = 3 because (HH)UHD,(H)UHD(H) and UHD(HH) are the only Motzkin paths of length 5 with 2 horizontal steps at level 0 (shown between parentheses); here U=(1,1), H=(1,0) and D=(1,-1).
Production matrix begins
  0, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 1, 1, 0, 1;
  1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 0, 1;
... - Philippe Deléham, Mar 02 2013

Maple

G:=2/(1-2*t*z+z+sqrt(1-2*z-3*z^2)): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..12);
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> A005043(n-1)); # Peter Luschny, Oct 09 2022

Mathematica

nmax = 12; t[n_, k_] := ((-1)^(n+k)*k*n!*HypergeometricPFQ[{(k+1)/2, k/2, k-n}, {k, k+1}, 4])/(n*k!*(n-k)!); Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)

Prog

(PARI) T(n, k) = ((k+1)/(n+1))*sum(j=k+1, n+1, (-1)^(n-j+1)*binomial(n+1, j)* binomial(2*j-k-2, j-1) ); \\ G. C. Greubel, Feb 18 2020
(Magma) [((k+1)/(n+1))*(&+[(-1)^(n-j+1)*Binomial(n+1, j)*Binomial(2*j-k-2, j-1): j in [k+1..n+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020
(Sage) [[((k+1)/(n+1))*sum( (-1)^(n-j+1)*binomial(n+1, j)* binomial(2*j-k-2, j-1) for j in (k+1..n+1)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 18 2020

Crossrefs

Cf. A001006, A005043, A187306.

Sequence in context: A091889 A147785 A067591 * A266692 A077884 A331103

Adjacent sequences: A097606 A097607 A097608 * A097610 A097611 A097612

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 30 2004

Status

approved