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%I #105 Feb 26 2023 08:26:50
%S 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,
%T 7,27,70,133,189,196,127,1,8,35,104,230,392,518,512,323,1,9,44,147,
%U 369,726,1140,1422,1353,835,1,10,54,200,560,1242,2235,3288,3915,3610,2188
%N 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 Right-hand columns have g.f. M^k, where M is g.f. of Motzkin numbers.
%C Consider a semi-infinite chessboard with squares labeled (n,k), ranks or rows n >= 0, files or columns k >= 0; number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - _Harrie Grondijs_, May 27 2005. Cf. A114929, A111808, A114972.
%D Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
%D A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
%H Reinhard Zumkeller, <a href="/A026300/b026300.txt">Rows n = 0..120 of triangle, flattened</a>
%H M. Aigner, <a href="http://dx.doi.org/10.1006/eujc.1998.0235">Motzkin Numbers</a>, Europ. J. Comb. 19 (1998), 663-675.
%H Tewodros Amdeberhan, Moa Apagodu, Doron Zeilberger, <a href="http://arxiv.org/abs/1507.07660">Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle</a>, arXiv:1507.07660 [math.CO], 2015.
%H J. L. Arregui, <a href="https://arxiv.org/abs/math/0109108">Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles</a>, arXiv:math/0109108 [math.NT], 2001.
%H F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.
%H H. Bottomley, <a href="/A001006/a001006.2.gif">Illustration of initial terms</a>
%H M. Buckley, R. Garner, S. Lack, R. Street, <a href="http://arxiv.org/abs/1307.0265">Skew-monoidal categories and the Catalan simplicial set</a>, arXiv preprint arXiv:1307.0265 [math.CT], 2013.
%H L. Carlitz, <a href="http://www.jstor.org/stable/2099558">Solution of certain recurrences</a>, SIAM J. Appl. Math., 17 (1969), 251-259.
%H J. L. Chandon, J. LeMaire and J. Pouget, <a href="http://www.numdam.org/item?id=MSH_1978__62__61_0">Denombrement des quasi-ordres sur un ensemble fini</a>, Math. Sci. Humaines, No. 62 (1978), 61-80.
%H Xiang-Ke Chang, X.-B. Hu, H. Lei, Y.-N. Yeh, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p8">Combinatorial proofs of addition formulas</a>, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, <a href="http://arxiv.org/abs/1507.04838">Idempotent Statistics of the Motzkin and Jones Monoids</a>, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2016.
%H Samuele Giraudo, <a href="https://arxiv.org/abs/1903.00677">Tree series and pattern avoidance in syntax trees</a>, arXiv:1903.00677 [math.CO], 2019.
%H Veronika Irvine, <a href="https://dspace.library.uvic.ca/handle/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016.
%H A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, <a href="http://dx.doi.org/10.1016/j.dam.2014.03.005">Complementary Riordan arrays</a>, Discrete Applied Mathematics, 172 (2014) 75-87.
%H A. Roshan, P. H. Jones and C. D. Greenman, <a href="http://arxiv.org/abs/1311.5769">An Exact, Time-Independent Approach to Clone Size Distributions in Normal and Mutated Cells</a>, arXiv preprint arXiv:1311.5769 [q-bio.QM], 2013.
%H M. János Uray, <a href="https://urayjanos.web.elte.hu/pub/expol-family.pdf">A family of barely expansive polynomials</a>, Eötvös Loránd University (Budapest, Hungary, 2020).
%H Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, <a href="http://www.cs.auckland.ac.nz/~mcw/Research/Outputs/Wils2013.pdf">PDF</a>.
%H Mark C. Wilson, <a href="http://dx.doi.org/10.1017/S0963548314000625">Diagonal asymptotics for products of combinatorial classes</a>, Combinatorics, Probability and Computing, Volume 24, Special Issue 01,January 2015, pp 354-372.
%H D. Yaqubi, M. Farrokhi D.G., H. Gahsemian Zoeram, <a href="https://arxiv.org/abs/1612.08697">Lattice paths inside a table. I</a>, arXiv:1612.08697 [math.CO], 2016-2017.
%F 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_, May 27 2004
%F T(n,k) = A027907(n,k) - A027907(n,k-2), k<=n.
%F Sum_{k=0..n} (-1)^k*T(n,k) = A099323(n+1). - _Philippe Deléham_, Mar 19 2007
%F Sum_{k=0..n} (T(n,k) mod 2) = A097357(n+1). - _Philippe Deléham_, Apr 28 2007
%F Sum_{k=0..n} T(n,k)*x^(n-k) = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -1, 0, 1, 2 respectively. - _Philippe Deléham_, Nov 28 2009
%F T(n,k) = binomial(n, k)*hypergeom([1/2 - k/2, -k/2], [n - k + 2], 4). - _Peter Luschny_, Mar 21 2018
%F T(n,k) = [t^(n-k)] [x^n] 2/(1 - (2*t + 1)*x + sqrt((1 + x)*(1 - 3*x))). - _Peter Luschny_, Oct 24 2018
%F The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + x + x^2)^n expanded about the point x = 0. - _Peter Bala_, Feb 26 2023
%e Triangle starts:
%e [0] 1;
%e [1] 1, 1;
%e [2] 1, 2, 2;
%e [3] 1, 3, 5, 4;
%e [4] 1, 4, 9, 12, 9;
%e [5] 1, 5, 14, 25, 30, 21;
%e [6] 1, 6, 20, 44, 69, 76, 51;
%e [7] 1, 7, 27, 70, 133, 189, 196, 127;
%e [8] 1, 8, 35, 104, 230, 392, 518, 512, 323;
%e [9] 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835.
%p A026300 := proc(n,k)
%p add(binomial(n,2*i+n-k)*(binomial(2*i+n-k,i) -binomial(2*i+n-k,i-1)), i=0..floor(k/2));
%p end proc: # _R. J. Mathar_, Jun 30 2013
%t t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}]; Table[ t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Robert G. Wilson v_, Jan 03 2011 *)
%t t[_, 0] = 1; t[n_, 1] := n; t[n_, k_] /; k>n || k<0 = 0; t[n_, n_] := t[n, n] = t[n-1, n-2]+t[n-1, n-1]; t[n_, k_] := t[n, k] = t[n-1, k-2]+t[n-1, k-1]+t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 18 2014 *)
%t T[n_, k_] := Binomial[n, k] Hypergeometric2F1[1/2 - k/2, -k/2, n - k + 2, 4];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Peter Luschny_, Mar 21 2018 *)
%o (Haskell)
%o a026300 n k = a026300_tabl !! n !! k
%o a026300_row n = a026300_tabl !! n
%o a026300_tabl = iterate (\row -> zipWith (+) ([0,0] ++ row) $
%o zipWith (+) ([0] ++ row) (row ++ [0])) [1]
%o -- _Reinhard Zumkeller_, Oct 09 2013
%o (PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(i=0, k\2, binomial(n, 2*i+n-k)*(binomial(2*i+n-k, i)-binomial(2*i+n-k, i-1))), ", ");); print(););} \\ _Michel Marcus_, Jul 25 2015
%Y Reflected version is in A064189.
%Y Row sums are in A005773.
%Y T(n,n) are Motzkin numbers A001006.
%Y Other columns of T include A002026, A005322, A005323.
%Y Cf. A099323, A097357, A005043, A059738, A027907, A020474, A059738.
%K nonn,tabl,nice
%O 0,5
%A _Clark Kimberling_
%E Corrected and edited by _Johannes W. Meijer_, Oct 05 2010
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