A132373 -id:A132373 - cp's OEIS frontend

A008315 Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 5, 1, 6, 14, 14, 1, 7, 20, 28, 14, 1, 8, 27, 48, 42, 1, 9, 35, 75, 90, 42, 1, 10, 44, 110, 165, 132, 1, 11, 54, 154, 275, 297, 132, 1, 12, 65, 208, 429, 572, 429, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 15, 104

Offset: 0

N. J. A. Sloane

There are several versions of a Catalan triangle: see A009766, A008315, A028364, A053121.
Number of standard tableaux of shape (n-k,k) (0<=k<=floor(n/2)). Example: T(4,1)=3 because in th top row we can have 124, 134, or 123 (but not 234). - Emeric Deutsch, May 23 2004
T(n,k) is the number of n-digit binary words (length n sequences on {0,1}) containing k 1's such that no initial segment of the sequence has more 1's than 0's. - Geoffrey Critzer, Jul 31 2009
T(n,k) is the number of dispersed Dyck paths (i.e. Motzkin paths with no (1,0) steps at positive heights) of length n and having k (1,1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, May 30 2011
T(n,k) is the number of length n left factors of Dyck paths having k (1,-1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), we have UUUUD, UUUDU, UUDUU, and UDUUU. There is a simple bijection between length n left factors of Dyck paths and dispersed Dyck paths of length n, that takes D steps into D steps. - Emeric Deutsch, Jun 19 2011
Triangle, with zeros omitted, given by (1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...) DELTA (0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
T(n,k) are rational multiples of A055151(n,k). - Peter Luschny, Oct 16 2015
T(2*n,n) = Sum_{k>=0} T(n,k)^2 = A000108(n), T(2*n+1,n) = A000108(n+1). - Michael Somos, Jun 08 2020

			Triangle begins:
  1;
  1;
  1, 1;
  1, 2;
  1, 3,  2;
  1, 4,  5;
  1, 5,  9,  5;
  1, 6, 14, 14;
  1, 7, 20, 28, 14;
  ...
T(5,2) = 5 because there are 5 such sequences: {0, 0, 0, 1, 1}, {0, 0, 1, 0, 1}, {0, 0, 1, 1, 0}, {0, 1, 0, 0, 1}, {0, 1, 0, 1, 0}. - _Geoffrey Critzer_, Jul 31 2009

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.

T. D. Noe, Rows n=0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015.
Nantel Bergeron, Kelvin Chan, Yohana Solomon, Farhad Soltani, and Mike Zabrocki, Quasisymmetric harmonics of the exterior algebra, arXiv:2206.02065 [math.CO], 2022.
Chassidy Bozeman, Christine Cheng, Pamela E. Harris, Stephen Lasinis, and Shanise Walker, The Pinnacle Sets of a Graph, arXiv:2406.19562 [math.CO], 2024. See pp. 9-10.
Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
Suyuong Choi and Younghan Yoon, A decomposition of graph a-numbers, arXiv:2508.06855 [math.CO], 2025. See p. 14.
C. Kenneth Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Lin Jiu, Victor H. Moll, and C. Vignat, Identities for generalized Euler polynomials, arXiv:1401.8037 [math.PR], 2014.
Nik Lygeros and Oliver Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Mustafa A. A. Obaid, S. Khalid Nauman, Wafaa M. Fakieh, and Claus Michael Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6.
Alon Regev, The central component of a triangulation, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.1, p. 7.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy]
Zheng Shi, Impurity entropy of junctions of multiple quantum wires, arXiv preprint arXiv:1602.00068 [cond-mat.str-el], 2016 (See Appendix A).
Index entries for sequences related to Chebyshev polynomials.

T(2n, n) = A000108 (Catalan numbers), row sums = A001405 (central binomial coefficients).
This is also the positive half of the triangle in A008482. - Michael Somos
This is another reading (by shallow diagonals) of the triangle A009766, i.e. A008315[n] = A009766[A056536[n]].
Cf. A120730, A055151.

a008315 n k = a008315_tabf !! n !! k
a008315_row n = a008315_tabf !! n
a008315_tabf = map reverse a008313_tabf
-- Reinhard Zumkeller, Nov 14 2013

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
T:= (n, k)-> b(n, n-2*k):
seq(seq(T(n, k), k=0..n/2), n=0..16);  # Alois P. Heinz, Oct 14 2022

Table[Binomial[k, i]*(k - 2 i + 1)/(k - i + 1), {k, 0, 20}, {i, 0, Floor[k/2]}] // Grid (* Geoffrey Critzer, Jul 31 2009 *)

{T(n, k) = if( k<0 || k>n\2, 0, if( n==0, 1, T(n-1, k-1) + T(n-1, k)))}; /* Michael Somos, Aug 17 1999 */

T(n, 0) = 1 if n >= 0; T(2*k, k) = T(2*k-1, k-1) if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) if k=1, 2, ..., floor(n/2). - Michael Somos, Aug 17 1999
T(n, k) = binomial(n, k) - binomial(n, k-1). - Michael Somos, Aug 17 1999
Rows of Catalan triangle A008313 read backwards. Sum_{k>=0} T(n, k)^2 = A000108(n); A000108 : Catalan numbers. - Philippe Deléham, Feb 15 2004
T(n,k) = C(n,k)*(n-2*k+1)/(n-k+1). - Geoffrey Critzer, Jul 31 2009
Sum_{k=0..n} T(n,k)*x^k = A000012(n), A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 12 2011

Expanded description from Clark Kimberling, Jun 15 1997

A120730 Another version of Catalan triangle A009766.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 5, 4, 1, 0, 0, 0, 5, 9, 5, 1, 0, 0, 0, 0, 14, 14, 6, 1, 0, 0, 0, 0, 14, 28, 20, 7, 1, 0, 0, 0, 0, 0, 42, 48, 27, 8, 1, 0, 0, 0, 0, 0, 42, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 0, 132, 165, 110, 44, 10, 1

Offset: 0

Philippe Deléham, Aug 17 2006, corrected Sep 15 2006

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, ...] DELTA [1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...] where DELTA is the operator defined in A084938.
Aerated version gives A165408. - Philippe Deléham, Sep 22 2009
T(n,k) is the number of length n left factors of Dyck paths having k up steps. Example: T(5,4)=4 because we have UDUUU, UUDUU, UUUDU, and UUUUD, where U=(1,1) and D=(1,-1). - Emeric Deutsch, Jun 19 2011
With zeros omitted: 1,1,1,1,2,1,2,3,1,5,4,1,... = A008313. - Philippe Deléham, Nov 02 2011

			As a triangle, this begins:
  1;
  0,  1;
  0,  1,  1;
  0,  0,  2,  1;
  0,  0,  2,  3,  1;
  0,  0,  0,  5,  4,  1;
  0,  0,  0,  5,  9,  5,  1;
  0,  0,  0,  0, 14, 14,  6,  1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000007, A000096, A000108, A001405, A001477, A003161, A005586, A005587.
Cf. A008313, A009766, A039598, A039599, A084938, A099376, A105523, A121725.
Cf. A126087, A128386, A121724, A128387, A129123, A132373, A132374, A132375.
Cf. A132889, A151162, A151254, A151281, A156195, A156361, A156362, A156566.
Cf. A156577, A165408, A357654.

A120730:= func< n,k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
[A120730(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Nov 07 2022

G := 4*z/((2*z-1+sqrt(1-4*z^2*t))*(1+sqrt(1-4*z^2*t))): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form  # Emeric Deutsch, Jun 19 2011
# second Maple program:
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
T:= (n, k)-> b(n, 2*k-n):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Oct 13 2022

b[x_, y_]:= b[x, y]= If[y<0 || y>x, 0, If[x==0, 1, Sum[b[x-1, y+j], {j, {-1, 1}}] ]];
T[n_, k_] := b[n, 2 k - n];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 21 2022, after Alois P. Heinz *)
T[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 07 2022 *)

def A120730(n,k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
flatten([[A120730(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 07 2022

G.f.: G(t,z) = 4*z/((2*z-1+sqrt(1-4*t*z^2))*(1+sqrt(1-4*t*z^2))). - Emeric Deutsch, Jun 19 2011
Sum_{k=0..n} x^k*T(n,n-k) = A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x=1,2,3,4,5,6,7,8,9 respectively. [corrected by Philippe Deléham, Oct 16 2008]
T(2*n,n) = A000108(n); A000108: Catalan numbers.
From Philippe Deléham, Oct 18 2008: (Start)
Sum_{k=0..n} T(n,k)^2 = A000108(n) and Sum_{n>=k} T(n,k) = A000108(k+1).
Sum_{k=0..n} T(n,k)^3 = A003161(n).
Sum_{k=0..n} T(n,k)^4 = A129123(n). (End)
Sum_{k=0..n}, T(n,k)*x^k = A000007(n), A001405(n), A151281(n), A151162(n), A151254(n), A156195(n), A156361(n), A156362(n), A156566(n), A156577(n) for x=0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Feb 10 2009
From G. C. Greubel, Nov 07 2022: (Start)
T(n, k) = 0 if n > 2*k, otherwise binomial(n, k)*(2*k-n+1)/(k+1).
Sum_{k=0..n} (-1)^k*T(n,k) = A105523(n).
Sum_{k=0..n} (-1)^k*T(n,k)^2 = -A132889(n), n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A357654(n).
T(n, n-1) = A001477(n).
T(n, n-2) = [n=2] + A000096(n-3), n >= 2.
T(n, n-3) = 2*[n<5] + A005586(n-5), n >= 3.
T(n, n-4) = 5*[n<7] - 2*[n=4] + A005587(n-7), n >= 4.
T(2*n+1, n+1) = A000108(n+1), n >= 0.
T(2*n-1, n+1) = A099376(n-1), n >= 1. (End)

cp's OEIS Frontend

A008315 Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

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