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 A055151 Triangular array of Motzkin polynomial coefficients. 21
 1, 1, 1, 1, 1, 3, 1, 6, 2, 1, 10, 10, 1, 15, 30, 5, 1, 21, 70, 35, 1, 28, 140, 140, 14, 1, 36, 252, 420, 126, 1, 45, 420, 1050, 630, 42, 1, 55, 660, 2310, 2310, 462, 1, 66, 990, 4620, 6930, 2772, 132, 1, 78, 1430, 8580, 18018, 12012, 1716, 1, 91, 2002, 15015, 42042 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS T(n,k) = number of Motzkin paths of length n with k up steps. T(n,k)=number of 0-1-2 trees with n edges and k+1 leaves, n>0. (A 0-1-2 tree is an ordered tree in which every vertex has at most two children.) E.g., T(4,1)=6 because we have UDHH, UHDH, UHHD, HHUD, HUHD, HUDH, where U=(1,1), D(1,-1), H(1,0). - Emeric Deutsch, Nov 30 2003 Coefficients in series reversion of x/(1+H*x+U*D*x^2) corresponding to Motzkin paths with H colors for H(1,0), U colors for U(1,1) and D colors for D(1,-1). - Paul Barry, May 16 2005 Eigenvector equals A119020, so that A119020(n) = Sum_{k=0..[n/2]} T(n,k)*A119020(k). - Paul D. Hanna, May 09 2006 Row reverse of A107131. - Peter Bala, May 07 2012 Also equals the number of 231-avoiding permutations of n+1 for which descents(w) = peaks(w) = k, where descents(w) is the number of positions i such that w[i]>w[i+1], and peaks(w) is the number of positions i such that w[i-1]w[i+1]. For example, T(4,1) = 6 because 13245, 12435, 14235, 12354, 12534, 15234 are the only 231-avoiding permutations of 5 elements with descents(w) = peaks(w) = 1. - Kyle Petersen, Aug 02 2013 Apparently, a refined irregular triangle related to this triangle (and A097610) is given in the Alexeev et al. link on p. 12. This entry's triangle is also related through Barry's comment to A125181 and A134264. The diagonals of this entry are the rows of A088617. - Tom Copeland, Jun 17 2015 The row length sequence of this irregular triangle is A008619(n) = 1 + floor(n/2). - Wolfdieter Lang, Aug 24 2015 REFERENCES Miklos Bona, Handbook of Enumerative Combinatorics, CRC Press (2015), page 617, Corollary 10.8.2 T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 4.3. LINKS Alois P. Heinz, Rows n = 0..200, flattened N. Alexeev, J. Andersen, R. Penner, and P. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, arXiv:1307.0967 [math.CO], 2013-2014. Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017. Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018. Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018. Colin Defant, Postorder Preimages, arXiv preprint arXiv:1604.01723 [math.CO], 2016. Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020. Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019. Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence. Paul W. Lapey and Aaron Williams, A Shift Gray Code for Fixed-Content Łukasiewicz Words, Williams College, 2022. Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013. E. Marberg, Actions and identities on set partitions, arXiv preprint arXiv:1107.4173 [math.CO], 2011-2012. MathOverflow, Motzkin polynomials and enumeration of chord diagrams. Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra, arXiv:2106.08257 [math.CO], 2021-2022. See p. 32. A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007. Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20. Claude Zeller and Robert Cordery, Light scattering as a Poisson process and first passage probability, arXiv:1906.11131 [cond-mat.stat-mech], 2019. FORMULA T(n,k) = n!/((n-2k)! k! (k+1)!) = A007318(n, 2k)*A000108(k). - Henry Bottomley, Jan 31 2003 E.g.f. row polynomials R(n,y): exp(x)*BesselI(1, 2*x*sqrt(y))/(x*sqrt(y)). - Vladeta Jovovic, Aug 20 2003 G.f. row polynomials R(n,y): 2 / (1 - x + sqrt((1 - x)^2 - 4 *y * x^2)). From Peter Bala, Oct 28 2008: (Start) The rows of this triangle are the gamma vectors of the n-dimensional (type A) associahedra (Postnikov et al., p. 38). Cf. A089627 and A101280. The row polynomials R(n,x) = Sum_{k = 0..n} T(n,k)*x^k begin R(0,x) = 1, R(1,x) = 1, R(2,x) = 1 + x, R(3,x) = 1 + 3*x. They are related to the Narayana polynomials N(n,x) := Sum_{k = 1..n} (1/n)*C(n,k)*C(n,k-1)*x^k through x*(1 + x)^n*R(n, x/(1 + x)^2) = N(n+1,x). For example, for n = 3, x*(1 + x)^3*(1 + 3*x/(1 + x)^2) = x + 6*x^2 + 6*x^3 + x^4, the 4th Narayana polynomial. Recursion relation: (n + 2)*R(n,x) = (2*n + 1)*R(n-1,x) - (n - 1)*(1 - 4*x)*R(n-2,x), R(0,x) = 1, R(1,x) = 1. (End) G.f.: M(x,y) satisfies: M(x,y)= 1 + x M(x,y) + y*x^2*M(x,y)^2. - Geoffrey Critzer, Feb 05 2014 T(n,k) = A161642(n,k)*A258820(n,k) = (binomial(n,k)/A003989(n+1, k+1))* A258820(n,k). - Tom Copeland, Jun 18 2015 Let T(n,k;q) = n!*(1+k)/((n-2*k)!*(1+k)!^2)*hypergeom([k,2*k-n],[k+2],q) then T(n,k;0) = A055151(n,k), T(n,k;1) = A008315(n,k) and T(n,k;-1) = A091156(n,k). - Peter Luschny, Oct 16 2015 From Tom Copeland, Jan 21 2016: (Start) Reversed rows of A107131 are rows of this entry, and the diagonals of A107131 are the columns of this entry. The diagonals of this entry are the rows of A088617. The antidiagonals (bottom to top) of A088617 are the rows of this entry. O.g.f.: [1-x-sqrt[(1-x)^2-4tx^2]]/(2tx^2), from the relation to A107131. Re-indexed and signed, this triangle gives the row polynomials of the compositional inverse of the shifted o.g.f. for the Fibonacci polynomials of A011973, x / [1-x-tx^2] = x + x^2 + (1+t) x^3 + (1+2t) x^4 + ... . (End) Row polynomials are P(n,x) = (1 + b.y)^n = Sum{k=0 to n} binomial(n,k) b(k) y^k = y^n M(n,1/y), where b(k) = A126120(k), y = sqrt(x), and M(n,y) are the Motzkin polynomials of A097610. - Tom Copeland, Jan 29 2016 Coefficients of the polynomials p(n,x) = hypergeom([(1-n)/2, -n/2], [2], 4x). - Peter Luschny, Jan 23 2018 Sum_{k=1..floor(n/2)} k * T(n,k) = A014531(n-1) for n>1. - Alois P. Heinz, Mar 29 2020 EXAMPLE The irregular triangle T(n,k) begins: n\k 0 1 2 3 4 5 ... 0: 1 1: 1 2: 1 1 3: 1 3 4: 1 6 2 5: 1 10 10 6: 1 15 30 5 7: 1 21 70 35 8: 1 28 140 140 14 9: 1 36 252 420 126 10: 1 45 420 1050 630 42 ... reformatted. - Wolfdieter Lang, Aug 24 2015 MAPLE b:= proc(x, y) option remember; `if`(y>x or y<0, 0, `if`(x=0, 1, expand( b(x-1, y) +b(x-1, y+1) +b(x-1, y-1)*t))) end: T:= n-> (p-> seq(coeff(p, t, i), i=0..degree(p)))(b(n, 0)): seq(T(n), n=0..20); # Alois P. Heinz, Feb 05 2014 MATHEMATICA m=(1-x-(1-2x+x^2-4x^2y)^(1/2))/(2x^2 y); Map[Select[#, #>0&]&, CoefficientList[ Series[m, {x, 0, 15}], {x, y}]]//Grid (* Geoffrey Critzer, Feb 05 2014 *) p[n_] := Hypergeometric2F1[(1-n)/2, -n/2, 2, 4 x]; Table[CoefficientList[p[n], x], {n, 0, 13}] // Flatten (* Peter Luschny, Jan 23 2018 *) PROG (PARI) {T(n, k) = if( k<0 || 2*k>n, 0, n! / ((n-2*k)! * k! * (k+1)!))} (PARI) {T(n, k) = if( k<0 || 2*k>n, 0, polcoeff( polcoeff( 2 / (1 - x + sqrt((1 - x)^2 - 4*y*x^2 + x * O(x^n))), n), k))} /* Michael Somos, Feb 14 2006 */ (PARI) {T(n, k) = n++; if( k<0 || 2*k>n, 0, polcoeff( polcoeff( serreverse( x / (1 + x + y*x^2) + x * O(x^n)), n), k))} /* Michael Somos, Feb 14 2006 */ CROSSREFS A107131 (row reversed), A080159 (with trailing zeros), A001006 = row sums, A000108(n) = T(2n, n), A001700(n) = T(2n+1, n), A119020 (eigenvector), A001263 (Narayana numbers), A089627 (gamma vectors of type B associahedra), A101280 (gamma vectors of type A permutohedra). Cf. A125181, A134264, A088617, A161642, A258820, A003989, A008315, A091156, A011973, A097610, A126120. Cf. A014531. Sequence in context: A270103 A134546 A152193 * A181187 A104573 A327438 Adjacent sequences: A055148 A055149 A055150 * A055152 A055153 A055154 KEYWORD nonn,tabf,easy AUTHOR Michael Somos, Jun 14 2000 STATUS approved

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