A026013 -id:A026013 - cp's OEIS frontend

A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 9, 1, 6, 15, 19, 9, 1, 7, 21, 34, 28, 1, 8, 28, 55, 62, 28, 1, 9, 36, 83, 117, 90, 1, 10, 45, 119, 200, 207, 90, 1, 11, 55, 164, 319, 407, 297, 1, 12, 66, 219, 483, 726, 704, 297, 1, 13, 78, 285, 702, 1209, 1430, 1001, 1, 14, 91, 363, 987, 1911, 2639, 2431, 1001

Offset: 0

Clark Kimberling

			From _Jonathon Kirkpatrick_, Jul 01 2016: (Start)
Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3;
  1,  4,  6,   3;
  1,  5, 10,   9;
  1,  6, 15,  19,   9;
  1,  7, 21,  34,  28;
  1,  8, 28,  55,  62,   28;
  1,  9, 36,  83, 117,   90;
  1, 10, 45, 119, 200,  207,   90;
  1, 11, 55, 164, 319,  407,  297;
  1, 12, 66, 219, 483,  726,  704,  297;
  1, 13, 78, 285, 702, 1209, 1430, 1001;
  ... (End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Diagonals of this sequence: A000217, A000245, A026012, A026013, A026014, A026015, A026016, A026017, A026018, A026019, A026020, A026021.

Sums involving this sequence: A026010, A027287, A027288, A027289, A027290, A027291, A027292.

[1] cat [Binomial(n,k) - Binomial(n,k-3): k in [0..Floor((n+2)/2)], n in [1..15]]; // G. C. Greubel, Mar 18 2021

T[n_, k_]:= Binomial[n, k] - Binomial[n, k-3];
Join[{1}, Table[T[n, k], {n,14}, {k,0,Floor[(n+2)/2]}]//Flatten] (* G. C. Greubel, Mar 18 2021 *)

[1]+flatten([[binomial(n,k) - binomial(n,k-3) for k in (0..(n+2)//2)] for n in (1..15)]) # G. C. Greubel, Mar 18 2021

T(n, k) = binomial(n, k) - binomial(n, k-3). - Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001

Sum_{k=0..floor((n+2)/2)} T(n, k) = A026010(n). - G. C. Greubel, Mar 18 2021

A375085 Triangle read by rows: T(n,k) is the number of ballotlike paths ending at (n, k), with 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 3, 2, 1, 5, 9, 6, 3, 1, 14, 28, 21, 10, 4, 1, 42, 90, 76, 39, 15, 5, 1, 132, 297, 276, 159, 64, 21, 6, 1, 429, 1001, 1002, 643, 288, 97, 28, 7, 1, 1430, 3432, 3641, 2555, 1281, 475, 139, 36, 8, 1, 4862, 11934, 13261, 10004, 5536, 2300, 733, 191, 45, 9, 1

Offset: 0

Stefano Spezia, Jul 29 2024

A ballotlike path is a lattice path in the 1st quadrant starting at (0, 0) and ending at (n, k) which uses the steps U = (1, 1), D = (1, -1), u = (1, 0) (for upstairs or umber) and d = (1, 0) (for downstairs or denim), subject to the conditions that the umber horizontal steps do not occur at height zero and the denim horizontal steps do not occur before the first down step. See pp. 8-10 in Lazar and Linusson.

			Triangle begins:
    0;
    0,   1;
    1,   1,   1;
    2,   3,   2,   1;
    5,   9,   6,   3,  1;
   14,  28,  21,  10,  4,  1;
   42,  90,  76,  39, 15,  5, 1;
  132, 297, 276, 159, 64, 21, 6, 1;
  ...

Alexander Lazar and Svante Linusson, Two-Row Set-Valued Tableaux: Catalan+k Combinatorics, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024) Article #80, 12 pp. See p. 10.

Cf. A000108, A026013, A057427 (diagonal), A071724, A375086 (row sums).

T[n_,k_]:=Binomial[2n-2,n-k-1]-Binomial[2n-2,n-k-2]+Binomial[n-2,n-k]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

from math import isqrt
from sympy import binomial
def A375085(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-binomial(a+1,2)
    return int(binomial(c:=a-1<<1,d:=a-b-1)-binomial(c,d-1)+binomial(a-2,d+1)) if n else 0 # Chai Wah Wu, Nov 14 2024

T(n,k) = binomial(2*n-2,n-k-1) - binomial(2*n-2,n-k-2) + binomial(n-2,n-k).
T(n,0) = A000108(n-1).
T(n,1) = A071724(n-1) for n > 0.
T(n+1,2) - T(n,2) = A026013(n-1) for n > 2.

A097608 Triangle read by rows: number of Dyck paths of semilength n and having abscissa of the leftmost valley equal to k (if no valley, then it is taken to be 2n; 2<=k<=2n).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 3, 3, 1, 1, 0, 1, 14, 9, 9, 4, 3, 1, 1, 0, 1, 42, 28, 28, 14, 10, 4, 3, 1, 1, 0, 1, 132, 90, 90, 48, 34, 15, 10, 4, 3, 1, 1, 0, 1, 429, 297, 297, 165, 117, 55, 35, 15, 10, 4, 3, 1, 1, 0, 1, 1430, 1001, 1001, 572, 407, 200, 125, 56, 35, 15, 10, 4, 3, 1, 1, 0, 1

Offset: 1

Emeric Deutsch, Aug 30 2004, Dec 22 2004

A valley point is a path vertex that is preceded by a downstep and followed by an upstep (or by nothing at all). T(n,k) is the number of Dyck n-paths whose first valley point is at position k, 2<=k<=2n. - David Callan, Mar 02 2005
Row n has 2n-1 terms.
Row sums give the Catalan numbers (A000108).
Columns k=2 through 7 are respectively A000108, A000245, A071724, A002057, A071725, A026013. The nonzero entries in the even-indexed columns approach A088218 and similarly the odd-indexed columns approach A001791.

			Triangle begins
\ k..2...3...4...5...6...7....
n
1 |..1
2 |..1...0...1
3 |..2...1...1...0...1
4 |..5...3...3...1...1...0...1
5 |.14...9...9...4...3...1...1...0...1
6 |.42..28..28..14..10...4...3...1...1...0...1
7 |132..90..90..48..34..15..10...4...3...1...1...0...1
T(4,3)=3 because we have UU(DU)DDUD, UU(DU)DUDD and UU(DU)UDDD, where U=(1,1), D=(1,-1) (the first valley, with abscissa 3, is shown between parentheses).

Cf. A000108, A000245.

G:=t^2*z*C*(1-t*z)/(1-t^2*z)/(1-t*z*C): C:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G,z=0,11)): for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: seq(seq(coeff(P[n],t^k),k=2..2*n),n=1..10);

G.f.=t^2*zC(1-tz)/[(1-t^2*z)(1-tzC)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

G.f. Sum_{2<=k<=2n}T(n, k)x^n*y^k = ((1 - (1 - 4*x)^(1/2))*y^2*(1 - x*y))/(2*(1 - ((1 - (1 - 4*x)^(1/2))*y)/2)*(1 - x*y^2)). With G:= (1 - (1 - 4*x)^(1/2))/2, the gf for column 2k is G(G^(2k+1)(G-x)-x^(k+1)(1-G))/(G^2-x) and for column 2k+1 is G(G-x)(G^(2k+2)-x^(k+1))/(G^2-x). - David Callan, Mar 02 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 23 2007

cp's OEIS Frontend

A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.

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A375085 Triangle read by rows: T(n,k) is the number of ballotlike paths ending at (n, k), with 0 <= k <= n.

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A097608 Triangle read by rows: number of Dyck paths of semilength n and having abscissa of the leftmost valley equal to k (if no valley, then it is taken to be 2n; 2<=k<=2n).

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