0,3

C(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0) and (0,1). - Joerg Arndt, Jul 01 2011

Row sums are A001700.

T(n, k) is also the number of order-preserving full transformations (of an n-chain) of waist k (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Oct 02 2008

If T(r,c), r=0,1,2,..., c=1,2,...,(r+1), are the triangle elements, then for r > 0, T(r,c) = binomial(r+c-1,c-1) = M(r,c) is the number of monotonic mappings from an ordered set of r elements into an ordered set of c elements. For example, there are 15 monotonic mappings from an ordered set of 4 elements into an ordered set of 3 elements. For c > r+1, use the identity M(r,c) = M(c-1,r+1) = T(c-1,r+1). For example, there are 210 monotonic mappings from an ordered set of 4 elements into an ordered set of 7 elements, because M(4,7) = T(6,5) = 210. Number of monotonic endomorphisms in a set of r elements, M(r,r), therefore appear on the second diagonal of the triangle which coincides with A001700. - Stanislav Sykora, May 26 2012

Start at the origin. Flip a fair coin to determine steps of (1,0) or (0,1). Stop when you are a (perpendicular) distance of n steps from the x axis or the y axis. For k = 0,1,...,n-1, C(n-1,k)/2^(n+k) is the probability that you will stop on the point (n,k). This is equal to the probability that you will stop on the point (k,n). Hence, Sum_{k=0..n} C(n,k)/2^(n+k) = 1. - Geoffrey Critzer, May 13 2017

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1)=1, arXiv:1909.11222 [math.NT], 2019. See Qn(x) Table 1 p. 2.

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)

A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.

A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.

T(n,k) = A092392(n,n-k), k = 0..n. - Reinhard Zumkeller, Jul 27 2012

T(n,k) = A178300(n,k), n>0, k = 1..n. - L. Edson Jeffery, Jul 23 2014

T(n,k) = (n + 1)*hypergeom([-n, 1 - k], [2], 1). - Peter Luschny, Jan 09 2022

T(n,k) = hypergeom([-n, -k], [1], 1). - Peter Luschny, Mar 21 2024

G.f.: 1/((1-2x*y*C(x*y))*(1-x*C(x*y))), where C(x) is the g.f. for A000108, the Catalan numbers. - Michael D. Weiner, Jul 31 2024

The triangle is the lower triangular part of the square array:

1| 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1, 2| 3, 4, 5, 6, 7, 8, 9, 10, ...

1, 3, 6| 10, 15, 21, 28, 36, 45, 55, ...

1, 4, 10, 20| 35, 56, 84, 120, 165, 220, ...

1, 5, 15, 35, 70| 126, 210, 330, 495, 715, ...

1, 6, 21, 56, 126, 252| 462, 792, 1287, 2002, ...

1, 7, 28, 84, 210, 462, 924| 1716, 3003, 5005, ...

1, 8, 36, 120, 330, 792, 1716, 3432| 6435, 11440, ...

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870| 24310, ...

1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620| ...

The array read by antidiagonals gives the binomial triangle.

From Reinhard Zumkeller, Jul 27 2012: (Start)

Take the first n elements of the n-th diagonal (NW to SE) of left half of Pascal's triangle and write it as n-th row on the triangle on the right side, see above

0: 1 1

1: 1 _ 1 2

2: 1 2 __ 1 3 6

3: 1 3 __ __ 1 4 10 20

4: 1 4 6 __ __ 1 5 15 35 70

5: 1 5 10 __ __ __ 1 6 21 56 .. ..

6: 1 6 15 20 __ __ __ 1 7 28 .. .. .. ..

7: 1 7 21 35 __ __ __ __ 1 8 .. .. .. .. .. ..

8: 1 8 28 56 70 __ __ __ __ 1 .. .. .. .. .. .. .. .. (End)

for n from 0 to 10 do seq( binomial(n+m, n), m = 0 .. n) od; # Zerinvary Lajos, Dec 09 2007

t[n_, k_] := Binomial[n + k, n]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2013 *)

(PARI) /* same as in A092566 but use */

steps=[[1, 0], [1, 0] ];

/* Joerg Arndt, Jul 01 2011 */

(Haskell)

import Data.List (transpose)

a046899 n k = a046899_tabl !! n !! k

a046899_row n = a046899_tabl !! n

a046899_tabl = zipWith take [1..] $ transpose a007318_tabl

-- Reinhard Zumkeller, Jul 27 2012

(Magma) /* As triangle */ [[Binomial(n+k, n): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 18 2015

(SageMath)

for n in (0..9):

print([multinomial(n, k) for k in (0..n)]) # Peter Luschny, Dec 24 2020

Cf. A000108, A046900, A001700, A007318, A034868, A092392, A239103, A178300.

Sequence in context: A078760 A348113 A103280 * A309220 A225632 A035206

Adjacent sequences: A046896 A046897 A046898 * A046900 A046901 A046902

nonn,tabl,easy,nice

More terms from James Sellers

approved