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%I #23 May 31 2016 02:40:28
%S 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,
%T 145,39,5,835,835,385,113,19,1,2188,2188,1030,322,64,6,5798,5798,2775,
%U 910,203,26,1,15511,15511,7525,2562,622,97,7,41835,41835,20526,7203
%N Triangle read by rows: counts Motzkin paths by length of final descent.
%C 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
%H A. Bernini, M. Bouvel and L. Ferrari, <a href="http://puma.dimai.unifi.it/18_3_4/BerniniBouvelFerrari.pdf">Some statistics on permutations avoiding generalized patterns</a>, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237.
%H Sen-Peng Eu, Shu-Chung Liu, Yeong-Nan Yeh, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00018-0">Taylor expansions for Catalan and Motzkin numbers</a>, Advances in Applied Mathematics 29, Issue 3 (2002), 345-357.
%H J. L. Jacobsen, and J. Salas, <a href="http://arxiv.org/abs/cond-mat/0407444">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions</a>, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444.
%F 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.
%e Triangle begins
%e 1;
%e 1;
%e 1, 1;
%e 2, 2;
%e 4, 4, 1;
%e 9, 9, 3;
%e 21, 21, 8, 1;
%e 51, 51, 21, 4;
%t 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}]
%Y Row sums are the Motzkin numbers (A001006), as are columns k=0 and k=1 (apart from initial 1's).
%K nonn,tabf
%O 0,5
%A _David Callan_, Oct 24 2004
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