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%I #85 Mar 21 2024 15:36:36
%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 and 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
%F T(n,k) = (n + 1)*hypergeom([-n, 1 - k], [2], 1)). - _Peter Luschny_, Jan 09 2022
%F T(n,k) = hypergeom([-n, -k], [1], 1). - _Peter Luschny_, Mar 21 2024
%e The triangle is the lower triangular part of the square array:
%e 1| 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2| 3, 4, 5, 6, 7, 8, 9, 10, ...
%e 1, 3, 6| 10, 15, 21, 28, 36, 45, 55, ...
%e 1, 4, 10, 20| 35, 56, 84, 120, 165, 220, ...
%e 1, 5, 15, 35, 70| 126, 210, 330, 495, 715, ...
%e 1, 6, 21, 56, 126, 252| 462, 792, 1287, 2002, ...
%e 1, 7, 28, 84, 210, 462, 924| 1716, 3003, 5005, ...
%e 1, 8, 36, 120, 330, 792, 1716, 3432| 6435, 11440, ...
%e 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870| 24310, ...
%e 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620| ...
%e The array read by antidiagonals gives the binomial triangle.
%e From _Reinhard Zumkeller_, Jul 27 2012: (Start)
%e 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
%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 .. .. .. .. .. .. .. .. (End)
%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 (Haskell)
%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
%o (SageMath)
%o for n in (0..9):
%o print([multinomial(n, k) for k in (0..n)]) # _Peter Luschny_, Dec 24 2020
%Y Cf. A046900, A001700, A007318, A034868, A092392, 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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