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Triangles Row Sum MotzkinSchroeder

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Motzkin numbers

row sum A001006

A034929 A triangle of Motzkin ballot numbers, read by rows.

A064191 Triangle T(n,k) (n >= 0, 0 <= k <= n) generalizing Motzkin numbers.

A080159 Triangular array of ways of drawing k non-intersecting chords between n points on a circle; i.e. Motzkin polynomial coefficients.

A091836 A triangle of Motzkin ballot numbers.

A097609 Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k horizontal steps at level 0.

A097610 Triangle read by rows: T(n,k) is number of Motzkin paths of length n and having k horizontal steps.

A097854 Triangle read by rows: T(n,k)=number of Motzkin paths of length n and having abscissa of first return (i.e. first down step hitting the x-axis) equal to k (k>0); T(n,0)=1 (accounts for the paths consisting only of level steps).

A104544 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k HH's, where H=(1,0).

A107131 A Motzkin related triangle.

A110470 Triangle, read by rows, where the g.f. of diagonal n, D_n(x) and the g.f. of row n-1, R_{n-1}(x), are related by: D_n(x) = R_{n-1}(x) / (1-x)^(n+1) for n>0 and the g.f. of the main diagonal is D_0(x) = 1/(1-x).

A128097 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k steps that touch the x-axis (1<=k<=n).

A143364 Triangle read by rows: T(n,k) is the number of {0-1-2}-trees with n edges and k protected vertices (0<=k<=n-1). A {0-1-2}-tree is an ordered tree in which the outdegree of every vertex is 0, 1, or 2. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

A144218 Eigentriangle, row sums and borders = offset variations of Motzkin numbers

A171380 Expansion of the first column of triangle T_(1,x), T(x,y) defined in A039599; T_(1,0)= A061554, T_(1,1)= A064189, T_(1,2)= A039599, T_(1,3)= A110877, T_(1,4)= A124576.

A204849 A Motzkin triangle by rows

A247286 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k weak peaks.


Schroeder numbers

row sum : A001003


A033282 Triangle read by rows: T(n,k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

A033877 Triangular array associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k<n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

A080245 Inverse of coordination sequence array A113413.

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

A084988 Numbers made with nonprime digits such that the sum of the digits is also not prime.

A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

A090985 Triangle read by rows: T(n,k) = number of dissections of a convex n-gon by nonintersecting diagonals, having exactly k triangles (n>=2, k>=0).

A091370 Triangle read by rows: T(n,k)=number of dissections of a convex n-gon by nonintersecting diagonals, having a k-gon over a fixed edge (base).

A106579 Triangular array associated with Schroeder numbers: T(0,0) = 1, T(n,0) = 0 for n>0; T(n,k) = 0 if k<n; T(n,k)=T(n,k-1)+T(n-1,k-1)+T(n-1,k).

A114656 Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; called also marked Dyck paths) of semilength n and having k peaks (1<=k<=n).

A114687 Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; called also marked Dyck paths) of semilength n and having k double rises (0<=k<=n-1).

A114706 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n and having k ascents (n>=1; 0<=k<=n-1). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. An ascent in a Schroeder path is a maximal sequence of consecutive U steps.

A114709 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k horizontal steps on the x-axis (0<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1.

A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

A126216 Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n>=1; 0<=k<=n-1).

A133336 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938 .

A250485 Regular triangle array: number of [0-r]-covering hierarchies with thickness = e.

A259099 Triangle read by rows: Kreweras's "Rule A_4 left thickness" numbers.


Schroeder "Big" numbers

row sum: A006318


A011117 Triangle of numbers S(x,y) = number of lattice paths from (0,0) to (x,y) that use step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x.

A033878 Triangular array associated with Schroeder numbers.

A060693 Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ....] where DELTA is the operator defined in A084938.

A088617 Triangle T(n,k) (n>=0, k=0..n) read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1).

A090981 Triangle read by rows: T(n,k)=number of Schroeder paths (i.e. lattice path in the first quadrant, from the origin to a point on the x-axis and consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) of length 2n and having k ascents (i.e. maximal strings of (1,1) steps).

A101275 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n having exactly k down steps hitting the x-axis.

A101281 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k low humps.

A101282 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k valleys.

A101894 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at odd height.

A101895 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at even height.

A101919 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at even heights.

A101920 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at an odd height.

A103136 Inverse of the Delannoy triangle.

A104219 Triangle read by rows: T(n,k) is number of Schroeder paths of length 2n and having k peaks at height 1 (0<=k<=n).

A104552 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n having trapezoid weight k.

A108891 Triangle read by rows: T(n,k) = number of Schroeder (or royal) n-paths (A006318) containing k returns to the diagonal y=x. (A northeast step lying on y=x contributes a return.)

A108916 Triangle of Schroeder paths counted by number of diagonal steps not preceded by an east step.

A110189 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Schroeder paths of length 2n, having k (1,0)-steps on the lines y=0 and y=1 (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).

A114655 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.

A132372 Triangle T(n,k), read by rows ; T(n,k) counts Schroeder n-paths whose ascent starting at the initial vertex has length k .

A144156 Eigentriangle, row sums = A006318

A145035 T(n,k) is the number of order-decreasing and order-preserving partial transformations (of an n-chain) of waist k (waist(alpha) = max(Im(alpha))).

A172040 Triangle T(n,k), read by rows, given by [0,1,2,1,2,1,2,1,2,1,2,...] DELTA [2,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

A175124 A symmetric triangle, with sum the large Schroeder numbers