A046899 Triangle in which n-th row is {binomial(n+k,k), k=0..n}, n >= 0.
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
Offset: 0
Examples
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)
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.
- Index entries for triangles and arrays related to Pascal's triangle
Programs
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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
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Magma
/* As triangle */ [[Binomial(n+k, n): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 18 2015
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Maple
for n from 0 to 10 do seq( binomial(n+m,n), m = 0 .. n) od; # Zerinvary Lajos, Dec 09 2007
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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 *) -
PARI
/* same as in A092566 but use */ steps=[[1, 0], [1, 0] ]; /* Joerg Arndt, Jul 01 2011 */
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SageMath
for n in (0..9): print([multinomial(n, k) for k in (0..n)]) # Peter Luschny, Dec 24 2020
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
T(n,k) = hypergeom([-n, -k], [1], 1). - Peter Luschny, Mar 21 2024
G.f.: 1/((1-2x*y*C(x*y))*(1-x*C(x*y))), where C(x) is the g.f. for A000108, the Catalan numbers. - Michael D. Weiner, Jul 31 2024
Extensions
More terms from James Sellers
Comments