A098979 Triangle read by rows: counts Motzkin paths by length of final descent.

1, 1, 1, 1, 2, 2, 4, 4, 1, 9, 9, 3, 21, 21, 8, 1, 51, 51, 21, 4, 127, 127, 55, 13, 1, 323, 323, 145, 39, 5, 835, 835, 385, 113, 19, 1, 2188, 2188, 1030, 322, 64, 6, 5798, 5798, 2775, 910, 203, 26, 1, 15511, 15511, 7525, 2562, 622, 97, 7, 41835, 41835, 20526, 7203

Offset

0,5

Comments

Apparently the rows of this entry are the antidiagonals of the transfer matrix of Table 3 on p. 40 of the Jacobsen and Salas link. - Tom Copeland, Dec 25 2015

Links

Table of n, a(n) for n=0..59.

A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237.

Sen-Peng Eu, Shu-Chung Liu, Yeong-Nan Yeh, Taylor expansions for Catalan and Motzkin numbers, Advances in Applied Mathematics 29, Issue 3 (2002), 345-357.

J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444.

Formula

G.f. ( (1 + x)*(1 - x*y) - (1 + x*y)(1 - 2*x - 3*x^2)^(1/2) )/( 2*x*(1 - y + x*y(1 + x*y)) ) = Sum_{n>=0, 1<=k<=n/2}T(n, k)x^n*y^k.

Example

Triangle begins
   1;
   1;
   1,  1;
   2,  2;
   4,  4,  1;
   9,  9,  3;
  21, 21,  8, 1;
  51, 51, 21, 4;

Mathematica

Clear[a] a[0, 0]=1; a[1, 0]=1; a[2, 0]=a[2, 1]=1; a[n_, r_]/; r<0 := 0; a[n_, r_]/; n>=2 && r==0:= Sum[a[n-1, n-1-j], {j, n-1}]; a[n_, r_]/; n>=3 && r >= 1 := a[n, r] = Sum[a[n-2, n-2-j], {j, n-r-1}]+Sum[a[n-1, n-1-j], {j, n-r-1}]; Table[a[n, r], {n, 0, 10}, {r, 0, n/2}]

Crossrefs

Row sums are the Motzkin numbers (A001006), as are columns k=0 and k=1 (apart from initial 1's).

Sequence in context: A071511 A119922 A097860 * A071928 A325445 A165207

Adjacent sequences: A098976 A098977 A098978 * A098980 A098981 A098982

Keyword

nonn,tabf

Author

David Callan, Oct 24 2004

Status

approved