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A046899 Triangle in which n-th row is {binomial(n+k,k), k=0..n}, n >= 0. 19
1, 1, 2, 1, 3, 6, 1, 4, 10, 20, 1, 5, 15, 35, 70, 1, 6, 21, 56, 126, 252, 1, 7, 28, 84, 210, 462, 924, 1, 8, 36, 120, 330, 792, 1716, 3432, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 1, 11, 66, 286, 1001 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

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

LINKS

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

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.

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

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

EXAMPLE

1

1, 2

1, 3, 6

1, 4, 10, 20

1, 5, 15, 35, 70

1, 6, 21, 56, 126, 252

1, 7, 28, 84, 210, 462, 924

1, 8, 36, 120, 330, 792, 1716, 3432

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870

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

1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756

.   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 .. .. .. .. .. .. .. .. .

- Reinhard Zumkeller, Jul 27 2012

MAPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A210237 A078760 A103280 * A225632 A035206 A210238

Adjacent sequences:  A046896 A046897 A046898 * A046900 A046901 A046902

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified November 15 03:35 EST 2018. Contains 317224 sequences. (Running on oeis4.)