%I A026300
%S A026300 1,1,1,1,2,2,1,3,5,4,1,4,9,12,9,1,5,14,25,30,21,1,6,20,44,69,76,51,1,7,
               27,
%T A026300 70,133,189,196,127,1,8,35,104,230,392,518,512,323,1,9,44,147,369,726,
               1140,
%U A026300 1422,1353,835,1,10,54,200,560,1242,2235,3288,3915,3610,2188
%N A026300 Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for 
               n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for 
               k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).
%C A026300 Right-hand columns have g.f. M^k, where M is g.f. of Motzkin numbers.
%C A026300 Consider a semi-infinite chessboard with squares labeled (i,j), i >= 
               0, j >= 0; number of king-paths of length j from (0,0) to (i,j), 
               0 <= i <= j, is T(j,i-j). - Harrie Grondijs (hgrondijs(AT)epo.org), 
               May 27 2005. Cf. A114929, A111808, A114972.
%D A026300 M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
%D A026300 F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 
               (1999) 73-112.
%D A026300 L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 
               (1969), 251-259.
%D A026300 J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres 
               sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
%D A026300 Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame 
               study-as-struggle.
%D A026300 A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans 
               in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
%H A026300 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent 
               and Bernoulli numbers</a> related to Motzkin and Catalan numbers 
               by means of numerical triangles.
%H A026300 H. Bottomley, <a href="a001006.2.gif">Illustration of initial terms</
               a>
%F A026300 T(n, k)=Sum(i=0, Floor(k/2), binomial(n, 2i+n-k)[binomial(2i+n-k, i)-binomial(2i+n-k, 
               i-1)]) - Herbert Kociemba (kociemba(AT)t-online.de), May 27 2004
%F A026300 T(n, k) = A027907(n, k) - A027907(n, k-2), k<=n.
%F A026300 Sum_{k, 0<=k<=n}(-1)^k*T(n,k)=A099323(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 19 2007
%F A026300 Sum(T(n,k) mod 2, 0<=k<=n) = A097357(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 28 2007
%F A026300 Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A005043(n), A001006(n), A005773(n+1), 
               A059738(n) for x = -1, 0, 1, 2 respectively. [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Nov 28 2009]
%e A026300 1; 1,1; 1,2,2; 1,3,5,4; 1,4,9,12,9; 1,5,14,25,30,21; ...
%Y A026300 Motzkin numbers (A001006) are T(n, n), other columns of T include A002026, 
               A005322, A005323.
%Y A026300 Cf. A020474.
%Y A026300 Row sums are in A005773.
%Y A026300 Reflected version is in A064189. Cf. A059738.
%Y A026300 Sequence in context: A054336 A079956 A140717 this_sequence A099514 A139687 
               A064581
%Y A026300 Adjacent sequences: A026297 A026298 A026299 this_sequence A026301 A026302 
               A026303
%K A026300 nonn,tabl,nice,new
%O A026300 0,5
%A A026300 Clark Kimberling (ck6(AT)evansville.edu)