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 A046899 Triangle in which n-th row is {binomial(n+k,k), k=0..n}, n >= 0. 22

%I

%S 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,

%T 462,924,1,8,36,120,330,792,1716,3432,1,9,45,165,495,1287,3003,6435,

%U 12870,1,10,55,220,715,2002,5005,11440,24310,48620,1,11,66,286,1001

%N Triangle in which n-th row is {binomial(n+k,k), k=0..n}, n >= 0.

%C 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

%C Row sums are A001700.

%C 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

%C 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

%C 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

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

%H Reinhard Zumkeller, <a href="/A046899/b046899.txt">Rows n=0..150 of triangle, flattened</a>

%H Karl Dilcher, Maciej Ulas, <a href="https://arxiv.org/abs/1909.11222">Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1)=1</a>, arXiv:1909.11222 [math.NT], 2019. See Qn(x) Table 1 p. 2.

%H H. W. Gould, <a href="/A007680/a007680.pdf">A class of binomial sums and a series transform</a>, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)

%H A. Laradji, and A. Umar, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.10.023">Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra 278, (2004), 342-359.

%H A. Laradji, and A. Umar, <a href="http://dx.doi.org/10.1007/s00233-005-0553-6">Combinatorial results for semigroups of order-preserving full transformations</a>, Semigroup Forum 72 (2006), 51-62.

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n,k) = A092392(n,n-k), k = 0..n. - _Reinhard Zumkeller_, Jul 27 2012

%F T(n,k) = A178300(n,k), n>0, k = 1..n. - _L. Edson Jeffery_, Jul 23 2014

%e 1

%e 1, 2

%e 1, 3, 6

%e 1, 4, 10, 20

%e 1, 5, 15, 35, 70

%e 1, 6, 21, 56, 126, 252

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

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

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

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

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

%e . Take the first n elements of the n-th diagonal (NW to SE) of left

%e . half of Pascal's triangle and write it as n-th row on the triangle

%e . on the right side, see above

%e . 0: 1 1

%e . 1: 1 _ 1 2

%e . 2: 1 2 __ 1 3 6

%e . 3: 1 3 __ __ 1 4 10 20

%e . 4: 1 4 6 __ __ 1 5 15 35 70

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

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

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

%e . 8: 1 8 28 56 70 __ __ __ __ 1 .. .. .. .. .. .. .. .. .

%e - _Reinhard Zumkeller_, Jul 27 2012

%p for n from 0 to 10 do seq( binomial(n+m,n), m = 0 .. n) od; # _Zerinvary Lajos_, Dec 09 2007

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

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

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

%o /* _Joerg Arndt_, Jul 01 2011 */

%o import Data.List (transpose)

%o a046899 n k = a046899_tabl !! n !! k

%o a046899_row n = a046899_tabl !! n

%o a046899_tabl = zipWith take [1..] \$ transpose a007318_tabl

%o -- _Reinhard Zumkeller_, Jul 27 2012

%o (MAGMA) /* As triangle */ [[Binomial(n+k, n): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 18 2015

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

%K nonn,tabl,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_

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Last modified February 23 16:19 EST 2020. Contains 332176 sequences. (Running on oeis4.)