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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, 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 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-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 CROSSREFS Cf. A046900, A001700, A007318, A034868, A239103, A178300. Sequence in context: A210237 A078760 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 January 24 10:22 EST 2020. Contains 331193 sequences. (Running on oeis4.)