VOOZH about

URL: https://oeis.org/A092392

⇱ A092392 - OEIS

login
A092392
Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.
30
1, 2, 1, 6, 3, 1, 20, 10, 4, 1, 70, 35, 15, 5, 1, 252, 126, 56, 21, 6, 1, 924, 462, 210, 84, 28, 7, 1, 3432, 1716, 792, 330, 120, 36, 8, 1, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1, 48620, 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1, 184756, 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1
OFFSET
0,2
COMMENTS
First column is C(2*n,n) or A000984. Central coefficients are C(3*n,n) or A005809. - Paul Barry, Oct 14 2009
T(n,k) = A046899(n,n-k), k = 0..n-1. - Reinhard Zumkeller, Jul 27 2012
From Peter Bala, Nov 03 2015: (Start)
Viewed as the square array [binomial (2*n + k, n + k)]n,k>=0 this is the generalized Riordan array ( 1/sqrt(1 - 4*x),c(x) ) in the sense of the Bala link, where c(x) is the o.g.f. for A000108.
The square array factorizes as ( 1/(2 - c(x)),x*c(x) ) * ( 1/(1 - x),1/(1 - x) ), which equals the matrix product of A100100 with the square Pascal matrix [binomial (n + k,k)]n,k>=0. See the example below. (End)
LINKS
Paul Barry, On the Central Coefficients of Riordan Matrices, J. Int. Seq. 16 (2013), #13.5.1.
Lanqi Du and Ethan Y. H. Li, Planar networks and total positivity of Riordan arrays, arXiv:2512.11419 [math.CO], 2025. See p. 10.
Juan B. Gil, Oscar A. Lopez, and Michael D. Weiner, On separable permutations and three other pairs in the Schröder class, arXiv:2603.25528 [math.CO], 2026. See pp. 2, 14.
Ik-Pyo Kim and Michael J. Tsatsomeros, Inverse Relations in Shapiro's Open Questions, arXiv:1707.06590 [math.CO], 2017. See p. 7.
FORMULA
As a number triangle, this is T(n, k) = if(k <= n, C(2*n - k, n), 0). Its row sums are C(2*n + 1, n + 1) = A001700. Its diagonal sums are A176287. - Paul Barry, Apr 23 2005
G.f. of column k: 2^k/[sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))^k].
As a number triangle, this is the Riordan array (1/sqrt(1 - 4*x), x*c(x)), c(x) the g.f. of A000108. - Paul Barry, Jun 24 2005
G.f.: A(x,y)=1/sqrt(1 - 4*x)/(1-y*x*C(x)), where C(x) is g.f. of Catalan numbers. - Vladimir Kruchinin, Mar 19 2016
EXAMPLE
From Paul Barry, Oct 14 2009: (Start)
Triangle begins
1,
2, 1,
6, 3, 1,
20, 10, 4, 1,
70, 35, 15, 5, 1,
252, 126, 56, 21, 6, 1,
924, 462, 210, 84, 28, 7, 1,
3432, 1716, 792, 330, 120, 36, 8, 1
Production array is
2, 1,
2, 1, 1,
2, 1, 1, 1,
2, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
As a square array = A100100 * square Pascal matrix:
/1 1 1 1 ...\ / 1 \/1 1 1 1 ...\
|2 3 4 5 ...| | 1 1 ||1 2 3 4 ...|
|6 10 15 21 ...| = | 3 2 1 ||1 3 6 10 ...|
|20 35 56 84 ...| |10 6 3 1 ||1 4 10 20 ...|
|70 ... | |35 ... ||1 ... |
- Peter Bala, Nov 03 2015
MAPLE
A092392 := proc(n, k)
binomial(2*n-k, n-k) ;
end proc:
seq(seq(A092392(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Feb 06 2015
MATHEMATICA
Table[Binomial[2 n - k, n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 19 2016 *)
PROG
(Haskell)
a092392 n k = a092392_tabl !! (n-1) !! (k-1)
a092392_row n = a092392_tabl !! (n-1)
a092392_tabl = map reverse a046899_tabl
-- Reinhard Zumkeller, Jul 27 2012
(Maxima)
C(x):=(1-sqrt(1-4*x))/2;
A(x, y):=(1/sqrt(1-4*x))/(1-y*C(x));
taylor(A(x, y), y, 0, 10, x, 0, 10); /* Vladimir Kruchinin, Mar 19 2016 */
(PARI) for(n=0, 10, for(k=0, n, print1(binomial(2*n - k, n), ", "))) \\ G. C. Greubel, Nov 22 2017
(Magma) /* As a triangle */ [[Binomial(2*n-k, n): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 22 2017
KEYWORD
nonn,tabl,easy,changed
AUTHOR
Ralf Stephan, Mar 21 2004
EXTENSIONS
Diagonal sums comment corrected by Paul Barry, Apr 14 2010
Offset corrected by R. J. Mathar, Feb 08 2013
STATUS
approved