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 A098979 Triangle read by rows: counts Motzkin paths by length of final descent. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified May 28 16:37 EDT 2022. Contains 354119 sequences. (Running on oeis4.)