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 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). 47
 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, 70, 133, 189, 196, 127, 1, 8, 35, 104, 230, 392, 518, 512, 323, 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835, 1, 10, 54, 200, 560, 1242, 2235, 3288, 3915, 3610, 2188 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Right-hand columns have g.f. M^k, where M is g.f. of Motzkin numbers. 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. REFERENCES Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle. A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147. LINKS Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675. Tewodros Amdeberhan, Moa Apagodu, Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015. J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001. F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112. H. Bottomley, Illustration of initial terms M. Buckley, R. Garner, S. Lack, R. Street, Skew-monoidal categories and the Catalan simplicial set, arXiv preprint arXiv:1307.0265 [math.CT], 2013. L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259. J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80. Xiang-Ke Chang, X.-B. Hu, H. Lei, Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8. I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2016. Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019. Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87. A. Roshan, P. H. Jones and C. D. Greenman, An Exact, Time-Independent Approach to Clone Size Distributions in Normal and Mutated Cells, arXiv preprint arXiv:1311.5769 [q-bio.QM], 2013. M. János Uray, A family of barely expansive polynomials, Eötvös Loránd University (Budapest, Hungary, 2020). Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, PDF. Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, Combinatorics, Probability and Computing, Volume 24, Special Issue 01,January 2015, pp 354-372. D. Yaqubi, M. Farrokhi D.G., H. Gahsemian Zoeram, Lattice paths inside a table. I, arXiv:1612.08697 [math.CO], 2016-2017. FORMULA 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 T(n,k) = A027907(n,k) - A027907(n,k-2), k<=n. Sum_{k=0..n} (-1)^k*T(n,k) = A099323(n+1). - Philippe Deléham, Mar 19 2007 Sum_{k=0..n} (T(n,k) mod 2) = A097357(n+1). - Philippe Deléham, Apr 28 2007 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 T(n,k) = binomial(n, k)*hypergeom([1/2 - k/2, -k/2], [n - k + 2], 4). - Peter Luschny, Mar 21 2018 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 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 EXAMPLE Triangle starts: [0] 1; [1] 1, 1; [2] 1, 2, 2; [3] 1, 3, 5, 4; [4] 1, 4, 9, 12, 9; [5] 1, 5, 14, 25, 30, 21; [6] 1, 6, 20, 44, 69, 76, 51; [7] 1, 7, 27, 70, 133, 189, 196, 127; [8] 1, 8, 35, 104, 230, 392, 518, 512, 323; [9] 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835. MAPLE A026300 := proc(n, k) 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)); end proc: # R. J. Mathar, Jun 30 2013 MATHEMATICA 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[_, 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[n_, k_] := Binomial[n, k] Hypergeometric2F1[1/2 - k/2, -k/2, n - k + 2, 4]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Mar 21 2018 *) PROG (Haskell) a026300 n k = a026300_tabl !! n !! k a026300_row n = a026300_tabl !! n a026300_tabl = iterate (\row -> zipWith (+) ([0, 0] ++ row) \$ zipWith (+) ([0] ++ row) (row ++ [0])) [1] -- Reinhard Zumkeller, Oct 09 2013 (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 CROSSREFS Reflected version is in A064189. Row sums are in A005773. T(n,n) are Motzkin numbers A001006. Other columns of T include A002026, A005322, A005323. Cf. A099323, A097357, A005043, A059738, A027907, A020474, A059738. Sequence in context: A140717 A257005 A160232 * A099514 A228352 A303911 Adjacent sequences: A026297 A026298 A026299 * A026301 A026302 A026303 KEYWORD nonn,tabl,nice AUTHOR Clark Kimberling EXTENSIONS Corrected and edited by Johannes W. Meijer, Oct 05 2010 STATUS approved

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