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A091836 A triangle of Motzkin ballot numbers. 3
1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 6, 6, 4, 1, 9, 13, 13, 10, 5, 1, 21, 30, 30, 24, 15, 6, 1, 51, 72, 72, 59, 40, 21, 7, 1, 127, 178, 178, 148, 105, 62, 28, 8, 1, 323, 450, 450, 378, 276, 174, 91, 36, 9, 1, 835, 1158, 1158, 980, 730, 480, 273, 128, 45, 10, 1, 2188, 3023, 3023 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Mirror image of A034929.

Column k has g.f.: z^k(1+zM)^(k+1).

T(n,0) = A086246(n+1) = A001006(n-1).

T(n,1) = A005554(n).

Row sums are the Motzkin numbers (A001006).

T(n-1,k) is the number of Motzkin paths of length n that have k points on the horizontal axis (besides the first and last point). For example T(1,0)=1 counts the path UD with 2 steps and no intermediate interception with the y=0 axis, and T(1,1)=1 counts the path FF with 2 steps, staying on the y=0 axis. - R. J. Mathar, Jul 23 2017

LINKS

Table of n, a(n) for n=0..68.

M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.

J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

FORMULA

G.f.: (1+zM)/(1-tz(1+zM)), where M = 1 + zM+ z ^2M^2 is the g.f. of the Motzkin numbers (A001006).

T(n,m) = (m*(Sum_{k=1..n-m} k*(-1)^(n+m+k)*binomial(n+k-1,n-1) * Sum_{j=0..n-m} binomial(j,-n+m-k+2*j)*binomial(n-m,j)))/(n*(n-m)), n>m, T(n,n)=1. - Vladimir Kruchinin, Aug 20 2012

EXAMPLE

Triangle begins:

   1;

   1,  1;

   1,  2,  1;

   2,  3,  3,  1;

   4,  6,  6,  4,  1;

   9, 13, 13, 10,  5,  1;

  21, 30, 30, 24, 15,  6,  1;

  ...

MATHEMATICA

T[n_, m_] := If[n == m, 1, (-1)^m (m Sum[k (-1)^(n+k) Binomial[n+k-1, n-1] Sum[Binomial[j, -n+m-k+2j] Binomial[n-m, j], {j, 0, n-m}], {k, 1, n-m}])/ (n(n-m))];

Table[T[n, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 27 2018, after Vladimir Kruchinin *)

PROG

(Maxima) T(n, m):=if n=m then 1 else (-1)^m*(m*sum(k*(-1)^(n+k)*binomial(n+k-1, n-1)*sum(binomial(j, -n+m-k+2*j)*binomial(n-m, j), j, 0, n-m), k, 1, n-m))/(n*(n-m)); /* Vladimir Kruchinin, Aug 20 2012 */

CROSSREFS

Cf. A001006, A005554, A034929, A086246.

Sequence in context: A099569 A191579 A097724 * A291980 A238281 A080850

Adjacent sequences:  A091833 A091834 A091835 * A091837 A091838 A091839

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Mar 09 2004

STATUS

approved

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Last modified May 27 06:24 EDT 2019. Contains 323599 sequences. (Running on oeis4.)