A089627 -id:A089627 - cp's OEIS frontend

A055151 Triangular array of Motzkin polynomial coefficients.

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

Offset: 0

Michael Somos, Jun 14 2000

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

			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

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.

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.

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.

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

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 *)

{T(n, k) = if( k<0 || 2*k>n, 0, n! / ((n-2*k)! * k! * (k+1)!))}

{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 */

{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 */

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

A377148 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 4, 14, 60, 225, 796, 2764, 9304, 30580, 98700, 313422, 981548, 3037473, 9301620, 28222000, 84927760, 253699285, 752863840, 2220831160, 6515581600, 19021079866, 55276625304, 159967084164, 461150383400, 1324652146775, 3792447499916, 10824189204014

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A051286, A182884, A377145, A377152, A377153, A377158, A377159.
Cf. A377149, A377150.
Cf. A089627.

[&+[Binomial(k+3,3)*Binomial(k, n-k)^2: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, May 12 2025

Table[Sum[Binomial[k+3,3]*Binomial[k, n-k]^2,{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)

a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=3, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (1-x-x^2) * ((1-x-x^2)^2 + 6*x^3) / ((1-x-x^2)^2 - 4*x^3)^(7/2).

A109187 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 6, 0, 12, 0, 1, 0, 30, 0, 20, 0, 1, 20, 0, 90, 0, 30, 0, 1, 0, 140, 0, 210, 0, 42, 0, 1, 70, 0, 560, 0, 420, 0, 56, 0, 1, 0, 630, 0, 1680, 0, 756, 0, 72, 0, 1, 252, 0, 3150, 0, 4200, 0, 1260, 0, 90, 0, 1, 0, 2772, 0, 11550, 0, 9240, 0, 1980, 0, 110, 0, 1

Offset: 0

Emeric Deutsch, Jun 21 2005

A Grand Motzkin path is a path in the half-plane x >= 0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).
From Peter Bala, Feb 11 2017: (Start)
Consider an infinite 1-dimensional integer lattice with an oriented self-loop at each vertex. Then T(n,k) equals the number of walks of length n from a vertex to itself having k loops. There is a bijection between such walks and Grand Motzkin paths which takes a right step and a left step on the lattice to an up step U and a down step D of a Grand Motzkin path respectively, and takes traversing a loop on the lattice to the horizontal step H. See A282252 for the corresponding triangle of walks on a 2-dimensional lattice with self-loops. (End)

			T(3,1)=6 because we have hud,hdu,udh,duh,uhd,dhu, where u=(1,1),d=(1,-1), h=(1,0).
Triangle begins:
n\k   [0]  [1]   [2]   [3]   [4]   [5]   [6]  [7]  [8]  [9] [10]
[0]    1;
[1]    0,   1;
[2]    2,   0,    1;
[3]    0,   6,    0,    1;
[4]    6,   0,   12,    0,    1;
[5]    0,  30,    0,   20,    0,    1;
[6]   20,   0,   90,    0,   30,    0,    1;
[7]    0, 140,    0,  210,    0,   42,    0,   1;
[8]   70,   0,  560,    0,  420,    0,   56,   0,   1;
[9]    0, 630,    0, 1680,    0,  756,    0,  72,   0,   1;
[10] 252,   0, 3150,    0, 4200,    0, 1260,   0,  90,   0,   1;
[11] ...
From _Peter Bala_, Feb 11 2017: (Start)
The infinitesimal generator begins
      0
      0    0
      2    0     0
      0    6     0     0
     -6    0    12     0     0
      0  -30     0    20     0   0
     80    0   -90     0    30   0   0
      0  560     0  -210     0  42   0  0
  -2310    0  2240     0  -420   0  56  0  0
  ....
and equals the generalized exponential Riordan array [log(Bessel_I(0,2x)),x], and so has integer entries. (End)

Gheorghe Coserea, Rows n = 0..100, flattened
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.

Diagonal of rational function R(x, y, t) = 1/(1 - (x^2 + t*x*y + y^2)) with respect to x,y, i.e., T(n,k) = [(xy)^n*t^k] R(x,y,t). For t=0..7 we have the diagonals: A126869(t=0, column 0), A002426(t=1, row sums), A000984(t=2), A026375(t=3), A081671(t=4), A098409(t=5), A098410(t=6), A104454(t=7).

Cf. A089627, A109188, A105868, A171128, A108666, A282252, A298608.

G:=1/sqrt((1-t*z)^2-4*z^2):Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od;
with(PolynomialTools): CL := p -> CoefficientList(simplify(p), x):
C := (n,x) -> binomial(2*n,n)*hypergeom([-n,-n],[-n+1/2],1/2-x/4):
seq(print(CL(C(n,x))), n=0..11); # Peter Luschny, Jan 23 2018

p[0] := 1; p[n_] := GegenbauerC[n, -n , -x/2];
Flatten[Table[CoefficientList[p[n], x], {n, 0, 11}]] (* Peter Luschny, Jan 23 2018 *)

T(n,k) = if ((n-k)%2, 0, binomial(n,k)*binomial(n-k, (n-k)/2));
concat(vector(12, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 06 2018

G.f.: 1/sqrt((1-tz)^2-4z^2).
Row sums yield the central trinomial coefficients (A002426).
T(2n+1, 0) = 0.
T(2n, 0) = binomial(2n,n) (A000984).
Sum_{k=0..n} k*T(n,k) = A109188(n).
Except for the order, same rows as those of A105868.
Column k has e.g.f. (x^k/k!)*Bessel_I(0,2x). - Paul Barry, Mar 11 2006
T(n,k) = binomial((n+k)/2,k)*binomial(n,(n+k)/2)*(1+(-1)^(n-k))/2. - Paul Barry, Sep 18 2007
Coefficient array of the polynomials P(n,x) = x^n*hypergeom([1/2-n/2,-n/2], [1], 4/x^2). - Paul Barry, Oct 04 2008
G.f.: 1/(1-xy-2x^2/(1-xy-x^2/(1-xy-x^2/(1-xy-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Paul Barry, Apr 21 2010: (Start)
Exponential Riordan array [Bessel_I(0,2x), x].
Coefficient array of the polynomials P(n,x) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k, k)*x^(n - 2k).
Diagonal sums are the aerated central Delannoy numbers (A001850 with interpolated zeros). (End)
From Peter Bala, Feb 11 2017: (Start)
T(n,k) = binomial(n,k)*binomial(n-k,floor((n-k)/2))*(1 + (-1)^(n-k))/2.
T(n,k) = (n/k) * T(n-1,k-1).
T(n,k) = the coefficient of H^k in the expansion of (H + U + 1/U)^n.
n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k) * binomial(2*k,k) * t^(n-2*k) = coefficient of x^n in the expansion of (1 + t*x + x^2)^n.
R(n,t) = Sum_{k = 0..n} binomial(n,k)*binomial(2*k,k)*(t - 2)^(n-k).
d/dt(R(n,t)) = n*R(n-1,t).
R(n,t) = (1/Pi) * Integral_{x = 0..Pi} (t + 2*cos(x))^n dx.
Moment representation on a finite interval: R(n,t) = 1/Pi * Integral_{x = t-2 .. t+2} x^n/sqrt((t + 2 - x)*(x - t + 2)) dx.
Recurrence: n*R(n,t) = t*(2*n - 1)*R(n-1,t) - (t^2 - 4)*(n - 1)*R(n-2,t) with R(0,t) = 1 and R(1,t) = t.
R(n,t) = A002426 (t = 1), A000984 (t = 2), A026375 (t = 3), A081671 (t = 4), A098409 (t = 5), A098410 (t = 6) and A104454(t = 7).
The zeros of the row polynomials appear to lie on the imaginary axis in the complex plane. Also, the zeros of R(n,t) and R(n+1,t) appear to interlace on the imaginary axis.
The polynomials R(n,1 + t) are the row polynomials of A171128. (End)
From Peter Luschny, Jan 23 2018: (Start)
These are the coefficients of the polynomials G(n, -n , -x/2) where G(n, a, x) denotes the n-th Gegenbauer polynomial.
These polynomials can also be expressed as C(n, x) = binomial(2*n,n)*hypergeom([-n, -n], [-n+1/2], 1/2-x/4). (End)

A377145 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 3, 9, 34, 111, 351, 1103, 3384, 10224, 30536, 90222, 264186, 767663, 2215623, 6356907, 18143300, 51540885, 145801395, 410888595, 1153964520, 3230723826, 9019081038, 25112021154, 69750583164, 193303849531, 534602071341, 1475644537323, 4065845732794

Offset: 0

Seiichi Manyama, Oct 17 2024

nonn

Cf. A051286, A182884, A377148, A377152, A377153, A377158, A377159.
Cf. A377146, A377147.
Cf. A089627.

a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: ((1-x-x^2)^2 + 2*x^3) / ((1-x-x^2)^2 - 4*x^3)^(5/2).

A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 5, 20, 95, 400, 1561, 5915, 21610, 76585, 265075, 898622, 2992235, 9810290, 31727815, 101379175, 320464280, 1003259080, 3113576320, 9586763720, 29305985800, 88997753446, 268642069750, 806394498200, 2408144329250, 7157177344225, 21177323087891

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..2357

Cf. A051286, A182884, A377145, A377148, A377153, A377158, A377159.

Cf. A001873, A089627.

f:= proc(n) local k; add(binomial(k+4,4)*binomial(k,n-k)^2,k=0..n) end proc:
map(f, [$0..50]); # Robert Israel, Dec 05 2024

a(n) = sum(k=0, n, binomial(k+4, 4)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=4, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..2} A089627(4,k) * (1-x-x^2)^(4-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(9/2).

A377153 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 6, 27, 140, 651, 2772, 11354, 44640, 169371, 624742, 2248575, 7922124, 27397937, 93214632, 312559200, 1034507696, 3384194616, 10954244952, 35118346760, 111602517096, 351819819414, 1100912299156, 3421515852834, 10566654790176, 32441857824859, 99060134392422

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Cf. A051286, A182884, A377145, A377148, A377152, A377158, A377159.

Cf. A001874, A089627.

a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=5, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..2} A089627(5,k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(11/2).

A377158 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 7, 35, 196, 994, 4578, 20118, 84540, 341397, 1335103, 5078227, 18852428, 68519920, 244413820, 857393700, 2963013816, 10102413972, 34025396580, 113329367816, 373642488044, 1220412680410, 3951964394642, 12695738508950, 40484919514284, 128216539026261

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Cf. A051286, A182884, A377145, A377148, A377152, A377153, A377159.

Cf. A089627.

a(n) = sum(k=0, n, binomial(k+6, 6)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=6, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..3} A089627(6,k) * (1-x-x^2)^(6-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(13/2).

A377159 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 8, 44, 264, 1446, 7152, 33516, 149688, 640233, 2642992, 10582220, 41249000, 157050660, 585621960, 2143442400, 7715164176, 27353809188, 95660348904, 330377130644, 1127996393656, 3810881349814, 12750188169312, 42276102419916, 139008143200536, 453526927536969

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Cf. A051286, A182884, A377145, A377148, A377152, A377153, A377158.

Cf. A089627.

a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=7, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..3} A089627(7,k) * (1-x-x^2)^(7-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(15/2).

A101280 Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 8, 1, 22, 16, 1, 52, 136, 1, 114, 720, 272, 1, 240, 3072, 3968, 1, 494, 11616, 34304, 7936, 1, 1004, 40776, 230144, 176896, 1, 2026, 136384, 1328336, 2265344, 353792, 1, 4072, 441568, 6949952, 21953408, 11184128, 1, 8166, 1398000, 33981760

Offset: 1

Don Knuth, Jan 28 2005

Row n has ceiling(n/2) terms.
The paper by Shapiro et al. explains why T(n,k) is the number of permutations of n elements having k peaks and with the further property that every rise (ascent) is immediately followed by a peak. [That is, the permutation p_1 ... p_n has the further property that (j > 1 and p_{j-1} < p_j) implies (j < n and p_j > p_{j+1}).] For example, the T(4,1)=8 permutations in the case n=4, k=1 are 1423, 2143, 2431, 3142, 3241, 3421, 4231, 4132.
A more elegant way to state this property: T(n,k) is the number of permutations of n objects with k descents such that every descent is a peak. The eight examples for n=4 and k=1 are now 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
The rows of this triangle are the gamma vectors of the n-dimensional (type A) permutohedra (Postnikov et al., p. 31). Cf. A055151 and A089627. - Peter Bala, Oct 28 2008

			Triangle begins:
  1;
  1,
  1,   2;
  1,   8,
  1,  22,  16;
  1,  52, 136,
  1, 114, 720, 272;
  ...
From _Peter Bala_, Jun 26 2012: (Start)
n = 4: the 9 weighted plane increasing 0-1-2 trees on 4 vertices are
..................................................................
..4...............................................................
..|...............................................................
..3..........4...4...............4...4...............3...3........
..|........./.....\............./.....\............./.....\.......
..2....2...3.......3...2...3...2.......2...3...4...2.......2...4..
..|.....\./.........\./.....\./.........\./.....\./.........\./...
..1...(t)1........(t)1....(t)1........(t)1....(t)1........(t)1....
..................................................................
..3...4...4...3...................................................
...\./.....\./....................................................
.(t)2....(t)2.....................................................
....|.......|.....................................................
....1.......1.....................................................
Hence R(4,t) = 1 + 8*t.
(End)

D. Foata and V. Strehl, "Euler numbers and variations of permutations", in Colloquio Internazionale sulle Teorie Combinatorie, Rome, September 1973, (Atti dei Convegni Lincei 17, Rome, 1976), 129.
Guoniu Han, Frédéric Jouhet, Jiang Zeng, Two new triangles of q-integers via q-Eulerian polynomials of type A and B, Ramanujan J (2013) 31:115-127, DOI 10.1007/s11139-012-9389-3
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 4.

Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
François Bergeron, Philippe Flajolet and Bruno Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
Leonard Carlitz and Richard Scoville, Generalized Eulerian numbers: combinatorial applications, Journal für die reine und angewandte Mathematik 265 (1974), 111.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
Diego Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
Dominique Foata and Marcel-Paul Schützenberger, Théorie géometrique des polynômes eulériens, arXiv:math/0508232 [math.CO], 2005; Lecture Notes in Math. 138 (1970), 81-83.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, On certain combinatorial expansions of descent polynomials and the change of grammars, arXiv:1802.02861 [math.CO], 2018.
Shi-Mei Ma, Jianfeng Wang, Guiying Yan, Jean Yeh, and Yeong-Nan Yeh, Symmetric decompositions and Euler-Stirling statistics on Stirling permutations, arXiv:2507.17667 [math.CO], 2025. See p. 5.
Shi-Mei Ma and Yeong-Nan Yeh, The alternating run polynomials of permutations, arXiv:1904.11437 [math.CO], 2019. See p. 4.
Alexander Postnikov, Victor Reiner, and Lauren Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007. [_Peter Bala_, Oct 28 2008]
Louis W. Shapiro, Wen-Jin Woan, and Seyoum Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.
Andrei K. Svinin, Somos-4 equation and related equations, arXiv:2307.05866 [math.CA], 2023. See p. 16.

The numbers 2^{n-1-k} T(n, k) form the array shown in A008303.
Cf. A055151, A089627. - Peter Bala, Oct 28 2008
Cf. A008292, A094503, A080635 (row sums).

T:=proc(n,k) if k<0 then 0 elif n=1 and k=0 then 1 elif k>floor((n-1)/2) then 0 else (k+1)*T(n-1,k)+(2*n-4*k)*T(n-1,k-1) fi end: for n from 1 to 13 do seq(T(n,k),k=0..floor((n-1)/2)) od; # yields sequence in triangular form # Emeric Deutsch, Aug 06 2005

t[, k?Negative] = 0; t[1, 0] = 1; t[n_, k_] /; k > (n-1)/2 = 0; t[n_, k_] := t[n, k] = (k+1)*t[n-1, k] + (2*n-4*k)*t[n-1, k-1]; Table[t[n, k], {n, 1, 13}, {k, 0, (n-1)/2}] // Flatten (* Jean-François Alcover, Nov 22 2012 *)

G.f.: Sum_{n>=1, k>=0} T(n, k) t^k z^n/n! = C(t)(2-C(t))/(exp^(-z sqrt(1-4t)) + 1 - C(t)) - C(t), where the sum on k is actually finite, running up to ceiling(n/2) - 1; here C(t) = (1 - sqrt(1-4t))/2t is the generating function for the Catalan numbers (A000108).
Sum_{k} Eulerian(n, k) x^k = Sum_{k} T(n, k) x^k (1+x)^(n-1-2k). E.g., 1 + 11x + 11x^2 + x^3 = (1+x)^3 + 8x(1+x).
From Peter Bala, Jun 26 2012: (Start)
T(n,k) = 2^k*A094503(n,k+1).
Let r(t) = sqrt(1 - 2*t) and w(t) = (1 - r(t))/(1 + r(t)). Define F(t,z) = r(t)*(1 + w(t)*exp(r(t)*z))/(1 - w(t)*exp(r(t)*z)) = 1 + t*z + t*z^2/2! + (t+t^2)*z^3/3! + (t+4*t^2)*z^4/4! + ...; F(t,z) is the e.g.f. for A094503. The e.g.f. for the present table is A(t,z) := (F(2*t,z) - 1)/(2*t) = z + z^2/2! + (1+2*t)*z^3/3! + (1+8*t)*z^4/4! + ....
The e.g.f. A(t,z) satisfies the autonomous partial differential equation dA/dz = 1 + A + t*A^2 with A(t,0) = 0. It follows that the inverse function A(t,z)^(-1) may be expressed as an integral: A(t,z)^(-1) = int {x = 0..z} 1/(1+x+t*x^2) dx.
Applying [Dominici, Theorem 4.1] to invert the integral gives the following method for calculating the row polynomials R(n,t) of the table: let f(t,x) = 1+x+t*x^2 and let D be the operator f(t,x)*d/dx. Then R(n+1,t) = D^n(f(t,x)) evaluated at x = 0.
By Bergeron et al., Theorem 1, the row polynomial R(n,t) is the generating function for rooted plane increasing 0-1-2 trees on n vertices, where the vertices of outdegree 2 have weight t and all other vertices have weight 1. An example is given below.
Row sums A080635.
(End)

More terms from Emeric Deutsch, Aug 06 2005

A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.

Original entry on oeis.org

1, -1, 1, 1, -2, 2, -1, 3, -6, 6, 1, -4, 12, -24, 6, -1, 5, -20, 60, -30, 30, 1, -6, 30, -120, 90, -180, 20, -1, 7, -42, 210, -210, 630, -140, 140, 1, -8, 56, -336, 420, -1680, 560, -1120, 70

Offset: 0

Peter Luschny, Aug 02 2009

Given T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!, let A(n,k) = abs(T(n,k)) be the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*q^k. Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the extended Motzkin numbers A189912. (See A089627 for the Motzkin numbers and A194586 for the complementary Motzkin numbers.)

			1
-1, 1
1, -2, 2
-1, 3, -6, 6
1, -4, 12, -24, 6
-1, 5, -20, 60, -30, 30
1, -6, 30, -120, 90, -180, 20
-1, 7, -42, 210, -210, 630, -140, 140
1, -8, 56, -336, 420, -1680, 560, -1120, 70

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, The lost Catalan numbers.

Row sums give A163650, row sums of absolute values give A163865.

Aerated versions A194586 (odd case) and A089627 (even case).

a := proc(n,k) (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)! end:
seq(print(seq(a(n,k),k=0..n)),n=0..8);

t[n_, k_] := (-1)^(n - k)*Floor[k/2]!^(-2)*n!/(n - k)!; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

for(n=0,10, for(k=0,n, print1((-1)^(n -k)*( (floor(k/2))! )^(-2)*(n!/(n - k)!), ", "))) \\ G. C. Greubel, Aug 01 2017

T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!.

E.g.f.: egf(x,y) = exp(-x)*BesselI(0,2*x*y)*(1+x*y).

cp's OEIS Frontend

A055151 Triangular array of Motzkin polynomial coefficients.

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A377148 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.

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A109187 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps.

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A377145 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(k,n-k)^2.

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A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.

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A377153 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

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A377158 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(k,n-k)^2.

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