# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a055151
Showing 1-1 of 1

%I A055151 #178 Feb 26 2024 11:00:58
%S A055151 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,
%T A055151 252,420,126,1,45,420,1050,630,42,1,55,660,2310,2310,462,1,66,990,
%U A055151 4620,6930,2772,132,1,78,1430,8580,18018,12012,1716,1,91,2002,15015,42042
%N A055151 Triangular array of Motzkin polynomial coefficients.
%C A055151 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
%C A055151 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
%C A055151 Eigenvector equals A119020, so that A119020(n) = Sum_{k=0..[n/2]} T(n,k)*A119020(k). - _Paul D. Hanna_, May 09 2006
%C A055151 Row reverse of A107131. - _Peter Bala_, May 07 2012
%C A055151 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]>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
%C A055151 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
%C A055151 The row length sequence of this irregular triangle is A008619(n) = 1 + floor(n/2). - _Wolfdieter Lang_, Aug 24 2015
%D A055151 Miklos Bona, Handbook of Enumerative Combinatorics, CRC Press (2015), page 617, Corollary 10.8.2
%D A055151 T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 4.3.
%H A055151 Alois P. Heinz, <a href="/A055151/b055151.txt">Rows n = 0..200, flattened</a>
%H A055151 N. Alexeev, J. Andersen, R. Penner, and P. Zograf, <a href="http://arxiv.org/abs/1307.0967">Enumeration of chord diagrams on many intervals and their non-orientable analogs</a>, arXiv:1307.0967 [math.CO], 2013-2014.
%H A055151 Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, <a href="https://arxiv.org/abs/1703.07262">Motzkin Numbers: an Operational Point of View</a>, arXiv:1703.07262 [math.CO], 2017.
%H A055151 Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.
%H A055151 Paul Barry, <a href="https://arxiv.org/abs/1805.02274">On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1805.02274 [math.CO], 2018.
%H A055151 Colin Defant, <a href="http://arxiv.org/abs/1604.01723">Postorder Preimages</a>, arXiv preprint arXiv:1604.01723 [math.CO], 2016.
%H A055151 Colin Defant, <a href="https://arxiv.org/abs/2004.11367">Troupes, Cumulants, and Stack-Sorting</a>, arXiv:2004.11367 [math.CO], 2020.
%H A055151 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 A055151 Thomas Grubb and Frederick Rajasekaran, <a href="https://arxiv.org/abs/2009.00650">Set Partition Patterns and the Dimension Index</a>, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.
%H A055151 Paul W. Lapey and Aaron Williams, <a href="https://www.researchgate.net/profile/Aaron-Williams/publication/360053030_A_Shift_Gray_Code_for_Fixed-Content_Lukasiewicz_Words/">A Shift Gray Code for Fixed-Content Łukasiewicz Words</a>, Williams College, 2022.
%H A055151 Shi-Mei Ma, <a href="http://arxiv.org/abs/1304.6654">On gamma-vectors and the derivatives of the tangent and secant functions</a>, arXiv:1304.6654 [math.CO], 2013.
%H A055151 E. Marberg, <a href="http://arxiv.org/abs/1107.4173">Actions and identities on set partitions</a>, arXiv preprint arXiv:1107.4173 [math.CO], 2011-2012.
%H A055151 MathOverflow, <a href="http://mathoverflow.net/questions/209729/motzkin-polynomials-and-enumeration-of-chord-diagrams">Motzkin polynomials and enumeration of chord diagrams</a>.
%H A055151 Jean-Christophe Novelli and Jean-Yves Thibon, <a href="https://arxiv.org/abs/2106.08257">Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra</a>, arXiv:2106.08257 [math.CO], 2021-2022. See p. 32.
%H A055151 A. Postnikov, V. Reiner, and L. Williams, <a href="http://arxiv.org/abs/math/0609184">Faces of generalized permutohedra</a>, arXiv:math/0609184 [math.CO], 2006-2007.
%H A055151 Tad White, <a href="https://arxiv.org/abs/2401.01462">Quota Trees</a>, arXiv:2401.01462 [math.CO], 2024. See p. 20.
%H A055151 Claude Zeller and Robert Cordery, <a href="https://arxiv.org/abs/1906.11131">Light scattering as a Poisson process and first passage probability</a>, arXiv:1906.11131 [cond-mat.stat-mech], 2019.
%F A055151 T(n,k) = n!/((n-2k)! k! (k+1)!) = A007318(n, 2k)*A000108(k). - _Henry Bottomley_, Jan 31 2003
%F A055151 E.g.f. row polynomials R(n,y): exp(x)*BesselI(1, 2*x*sqrt(y))/(x*sqrt(y)). - _Vladeta Jovovic_, Aug 20 2003
%F A055151 G.f. row polynomials R(n,y): 2 / (1 - x + sqrt((1 - x)^2 - 4 *y * x^2)).
%F A055151 From _Peter Bala_, Oct 28 2008: (Start)
%F A055151 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.
%F A055151 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.
%F A055151 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)
%F A055151 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
%F A055151 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
%F A055151 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
%F A055151 From _Tom Copeland_, Jan 21 2016: (Start)
%F A055151 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.
%F A055151 O.g.f.: [1-x-sqrt[(1-x)^2-4tx^2]]/(2tx^2), from the relation to A107131.
%F A055151 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)
%F A055151 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
%F A055151 Coefficients of the polynomials p(n,x) = hypergeom([(1-n)/2, -n/2], [2], 4x). - _Peter Luschny_, Jan 23 2018
%F A055151 Sum_{k=1..floor(n/2)} k * T(n,k) = A014531(n-1) for n>1. - _Alois P. Heinz_, Mar 29 2020
%e A055151 The irregular triangle T(n,k) begins:
%e A055151 n\k 0  1   2    3   4  5 ...
%e A055151 0:  1
%e A055151 1:  1
%e A055151 2:  1  1
%e A055151 3:  1  3
%e A055151 4:  1  6   2
%e A055151 5:  1 10  10
%e A055151 6:  1 15  30    5
%e A055151 7:  1 21  70   35
%e A055151 8:  1 28 140  140  14
%e A055151 9:  1 36 252  420 126
%e A055151 10: 1 45 420 1050 630 42
%e A055151 ... reformatted. - _Wolfdieter Lang_, Aug 24 2015
%p A055151 b:= proc(x, y) option remember;
%p A055151       `if`(y>x or y<0, 0, `if`(x=0, 1, expand(
%p A055151        b(x-1, y) +b(x-1, y+1) +b(x-1, y-1)*t)))
%p A055151     end:
%p A055151 T:= n-> (p-> seq(coeff(p, t, i), i=0..degree(p)))(b(n, 0)):
%p A055151 seq(T(n), n=0..20);  # _Alois P. Heinz_, Feb 05 2014
%t A055151 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 *)
%t A055151 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 *)
%o A055151 (PARI) {T(n, k) = if( k<0 || 2*k>n, 0, n! / ((n-2*k)! * k! * (k+1)!))}
%o A055151 (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 */
%o A055151 (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 */
%Y A055151 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).
%Y A055151 Cf. A125181, A134264, A088617, A161642, A258820, A003989, A008315, A091156, A011973, A097610, A126120.
%Y A055151 Cf. A014531.
%K A055151 nonn,tabf,easy
%O A055151 0,6
%A A055151 _Michael Somos_, Jun 14 2000

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE