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 A046899 Triangle in which n-th row is {binomial(n+k,k), k=0..n}, n >= 0. 22
 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 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. 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 T(n,k) = (n + 1)*hypergeom([-n, 1 - k], [2], 1)). - Peter Luschny, Jan 09 2022 EXAMPLE 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) 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-François 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 (SageMath) for n in (0..9):     print([multinomial(n, k) for k in (0..n)]) # Peter Luschny, Dec 24 2020 CROSSREFS Cf. A046900, A001700, A007318, A034868, A092392, A239103, A178300. Sequence in context: A078760 A348113 A103280 * A309220 A225632 A035206 Adjacent sequences:  A046896 A046897 A046898 * A046900 A046901 A046902 KEYWORD nonn,tabl,easy,nice AUTHOR EXTENSIONS More terms from James A. Sellers STATUS approved

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Last modified August 13 16:07 EDT 2022. Contains 356107 sequences. (Running on oeis4.)