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Triangles Row Sum CatalanNumbers
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Catalan Numbers
A001263 Triangle of Narayana numbers T(n,k) = C(n-1,k-1)C(n,k-1)/k with 1<=k<=n, read by rows. Also called the Catalan triangle.
A009766 Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).
A033184 Catalan triangle A009766 transposed.
A046726 Triangle of numbers of semi-meanders of order n with k components.
A047812 Parker's partition triangle read by rows.
A059365 Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1)-binomial(2*r-s-1,r), r>=0, 0<=s<=r.
A065600 Triangle T(n,k) giving number of Dyck paths of length 2n with exactly k hills (0 <= k <= n).
A073429 Upper triangular region of the table A073345.
A073430 Upper triangular region of the table A073346.
A078391 Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).
A080936 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and height k (1<=k<=n).
A090181 Triangle of Narayana (A001263) with 0<=k<=n, read by rows.
A091187 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches.
A091866 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having pyramid weight k.
A091867 Triangle read by rows: T(n,k)=number of Dyck paths of semilength n having k peaks at odd height.
A091869 Triangle read by rows: T(n,k)=number of Dyck paths of semilength n having k peaks at even height.
A094322 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k pyramids.
A094449 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having sum of pyramid heights equal to k.
A094507 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k UDUD's (here U=(1,1), D=(1,-1)).
A096793 Triangle read by rows: a(n,k) is the number of Dyck n-paths containing k odd-length ascents.
A096794 Triangle read by rows: a(n,k) = number of Dyck n-paths such that number of DUs at level 1 plus number of UDs at level 2 is k, 0<=k<=n-1.
A098747 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level.
A098977 Triangle read by rows: counts ordered trees by number of edges and position of first edge that terminates at a vertex of outdegree 1.
A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.
A100537 Triangle read by rows: T(n,k) is the number of Dyck n-paths whose first descent has length k.
A101276 Triangle read by rows: T(n,k) is the number of ordered trees having n edges and k branches of length 1.
A101975 Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0<= k <n).
A102003 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of odd length (n>=0, 0<=k<=n).
A102404 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n having k ascents of length 1 starting at an even level.
A106566 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, . . . ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, . . . ] where DELTA is the operator defined in A084938.
A108746 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k peaks that are not of the form uudd (here u=(1,1), d=(1,-1)).
A108838 Triangle of Dyck paths counted by number of long interior inclines.
A112307 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n-1).
A112413 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and starting with exactly k UD's, where U=(1,1), D=(1,-1) (0<=k<=n).
A114276 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having length of second ascent equal to k (0<=k<=n-1).
A120059 Triangle read by rows: T(n,k) is the number of Dyck n-paths (A000108) whose longest pyramid has size k.
A120060 Triangle read by rows: T(n,k) is the number of Dyck n-paths (A000108) whose longest sawtooth has size k.
A120988 Triangle read by rows: T(n,k) is the number of binary trees with n edges and such that the first leaf in the preorder traversal is at level k (1<=k<=n). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.
A121448 Triangle read by rows: T(n,k) is the number of binary trees with n edges and having k vertices of outdegree 1 (n>=0, k>=0). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.
A121685 Triangle read by rows: T(n,k) is the number of binary trees having n edges and k branches (1<=k<=n). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.
A123880 Inverse of number triangle A123878.
A124926 Triangle read by rows: T(n,k)=binom(n,k)*r(k), where r(k) are the Riordan numbers (r(k)=A005043(k); 0<=k<=n).
A126217 Triangle read by rows: T(n,k) is the number of 321-avoiding permutations of {1,2,...,n} having longest increasing subsequence of length k (1<=k<=n).
A126222 Triangle read by rows: T(n,k) is the number of 2-Motzkin paths (i.e. Motzkin paths with blue and red level steps) without red level steps on the x-axis, having length n and k level steps (0<=k<=n).
A127153 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k UDUD's starting at level 0; here U=(1,1), D=(1,-1) (0<=k<=n-1).
A127156 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D=(1,-1). This definition differs from the one in A091866.
A127158 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of length 1 starting from the root (0<=k<=n).
A127538 Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k odd-length branches starting at the root (0<=k<=n).
A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .
A131198 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,0,1,0,1,0,1,0,...] DELTA [0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938 .
A131427 A000108(n) preceded by n zeros.
A136621 Transpose Parker's partition triangle (A047812).
A141058 Pats by first entry.
A143362 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k protected vertices (0<=k<=n-1). A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.
A143949 Triangle read by rows: T(n,k) is the number of n-Dyck paths containing k odd-length descents to ground level (0<=k<=n).
A143953 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peaks in their peak plateaux (0<=k<=n-1). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.
A152879 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peaks of maximum height (1<=k<=n).
A154559 Triangle read by rows, A007318 * (A129186 * (A001006 * 0^(n-k))
A166073 Triangle read by rows: a(n,k)=number of permutations in S_n which avoid the pattern 123 and have exactly k descents.
A168511 Triangle T(n,k), read by rows, given by [0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,...] DELTA [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,...] where DELTA is the operator defined in A084938.
A171567 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A168491.
A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.
A178518 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} having genus 0 and such that p(1)=k (see first comment for definition of genus).
A181645 Triangle Id-(xc(x),xc(x)), c(x) the g.f. of the Catalan numbers A000108.
A196182 Triangle T(n,k), read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.
A203717 A Catalan triangle by rows.
A213600 Triangle T(n,k) read by rows: Number of Dyck n-paths with midpoint at height k.
A213946 A Catalan triangle read by rows, derived from the INVERT transform of initial segments of the Catalan numbers A000108.
A216948 Triangle read by rows: coefficients in expansion of number of r-colored set partitions of n as a polynomial in r.
A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.
A244530 Number T(n,k) of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
A247285 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n (n>=1) having k (0<=k<=n-1) upper interactions.
A259824 A Catalan-type triangle read by rows, generated by iteration of convolution squares.
A273821 Triangle read by rows: T(n,k) is the number of 123-avoiding permutations p of [n] (A000108) such that k is maximal with the property that the k largest entries of p, taken in order, avoid 132.
Catalan Numbers -1
A033184 Catalan triangle A009766 transposed.
A059365 Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1)-binomial(2*r-s-1,r), r>=0, 0<=s<=r.
A065600 Triangle T(n,k) giving number of Dyck paths of length 2n with exactly k hills (0 <= k <= n).
A091869 Triangle read by rows: T(n,k)=number of Dyck paths of semilength n having k peaks at even height.
A094322 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k pyramids.
A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.
A102404 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n having k ascents of length 1 starting at an even level.
A106566 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, . . . ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, . . . ] where DELTA is the operator defined in A084938.
A112307 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n-1).
A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).
A127156 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D=(1,-1). This definition differs from the one in A091866.
A171567 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A168491.
A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.
A196182 Triangle T(n,k), read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.
A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.
Catalan(n)+1
A181645 Triangle Id-(xc(x),xc(x)), c(x) the g.f. of the Catalan numbers A000108.
Catalan(2n)
none found yet
Catalan(2n+1)
none found yet
2*Catalan(n)
A156006 A symmetrical triangle based on A009799: a(n,m) = -(m - n)/(m + n)*Binomial[n + m, n]; t(n,m) = If[n == 0, 1, a(n, m) + a(n, n - m)]
A156197 A symmetrical triangle sequence: t(n,m)=-Binomial[m + n, -1 + m] + Binomial[m + n, n] + Binomial[ -m + 2 n, n] - Binomial[ -m + 2 n, -1 - m + n].
n*Catalan(n) = Binomial(2n, n-1) = A001791
A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.
A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).
A062196 Coefficient triangle of certain polynomials N(2; m,x).
A128821 Triangle T(n,k), 1<=k<=n, read by rows defined by :T(n,k)=C(n,k)*C(n-1,k-1)+C(n,k-1)*C(n-1,k)where C(n,k)=A007318(n,k) .
A132812 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
A136534 A001263 * A128064 (unsigned).
A136536 A001263 * A128064 * A000012 as infinite lower triangular matrices.
A141811 Partial Catalan numbers: triangle read by rows n = 1, 2, 3, ... and columns k = 0, 1, ..., n-1.
A172417 n*Catalan number(n+1) triangle.
A176992 Triangle T(n,m) = binomial(2n-k+1,n+1) read by rows, 0<=k<=n.
A178300 Triangle T(n,k) = binomial(n+k-1,n) read by rows, 1<=k<=n.
(n+1)*Catalan(n) = Binomial(2n, n) = A000984
A008459 Square the entries of Pascal's triangle.
A027555 Triangle of binomial coefficients C(-n,k).
A039599 Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).
A050165 Triangle read by rows: T(n,k)=M(2n+1,k,-1), 0<=k<=n, n >= 0, array M as in A050144.
A059481 Triangle T(n,k) = binomial(n+k-1,k), 0 <= k <= n, read by rows.
A097692 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs.
A100100 Triangle T(n,k) = binomial(2*n-k-1, n-k) read by rows.
A108747 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).
A118920 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross the x-axis k times (n>=1, k>=0).
A118921 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n having first return to the x-axis at (2k,0) (n,k>=1). (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).
A118963 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises (n>=1,k>=0).
A118964 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises above the x-axis (n>=1,k>=0). (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1); a double rise in a Grand Dyck path is an occurrence of uu in the path.)
A135091 A007318 * triangle M, where M = A002426 * 0^(n-k), 0<=k<=n.
A152229 Eigentriangle, row sums = A000984
A155788 Renewal array for 1/(x+sqrt(1-4x)).
A158815 Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.
A168256 Triangle read by rows: Catalan number C(n) repeated n+1 times.
A171128 A117852*A130595 as lower triangular matrices.
A171150 Triangle related to T(x,2x).
A171698 Square 'central binomial coefficients'-array read by anti-diagonals
A171824 Triangle T(n,k)= binomial(n + k,n) + binomial(2*n-k,n) read by rows.
A205946 Triangle read by rows related to A000984, central binomial coefficients
A212207 Triangle read by rows: coefficients of polynomials p_{n,n-1}(x) arising in enumeration of two-line arrays.
A213745 Triangle of numbers C^(6)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 6 appearances allowed.
A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed.
A227924 Triangle T(n,k): the number of binary sequences of n zeros and n ones in which the shortest run is of length k.
A229756 Triangle T(n,k): the number of binary sequences of n zeros and n ones in which the longest run is of length k.
A274293 Triangle read by rows: T(n,k) (0 <= k <= n) given by T(n,0) = 1, T(n,1) = 2^n - 1, T(n,2) = 2^n - 2, T(n,n-1) = T(n,n) = binomial(2n-2,n-1); and the other internal entries satisfy T(n,k) = T(n,k-1) + T(n-1,k).
n! * Catalan(n) = A001761 = (2n)!/(n+1)!
A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.
A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).
A062196 Coefficient triangle of certain polynomials N(2; m,x).
A128821 Triangle T(n,k), 1<=k<=n, read by rows defined by :T(n,k)=C(n,k)*C(n-1,k-1)+C(n,k-1)*C(n-1,k)where C(n,k)=A007318(n,k) .
A132812 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
A136534 A001263 * A128064 (unsigned).
A136536 A001263 * A128064 * A000012 as infinite lower triangular matrices.
A141811 Partial Catalan numbers: triangle read by rows n = 1, 2, 3, ... and columns k = 0, 1, ..., n-1.
A172417 n*Catalan number(n+1) triangle.
A176992 Triangle T(n,m) = binomial(2n-k+1,n+1) read by rows, 0<=k<=n.
A178300 Triangle T(n,k) = binomial(n+k-1,n) read by rows, 1<=k<=n.
(n+1)! * Catalan(n) = A001813 = (2n)!/(n)!
A038455 A Jabotinsky-triangle related to A006963.
A064307 Triangle of coefficients of certain numerator polynomials N(n,x).
A156653 Coefficients of a higher level infinite sum polynomial: p(x,n)=(1 - x)^(2n + 1)/((n + 1)*x^n)*Sum[(k + 1)^n*Binomial[k, n]*x^ k, {k, 0, Infinity}].
A220883 Triangle read by rows: row n gives coefficients of expansion of Product_{k = 1..n-1} ((n + 1)*x + k), starting with lowest power.
A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.
