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A092392 Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n. 25
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 (list; table; graph; refs; listen; history; text; internal format)
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

Reinhard Zumkeller, Rows n=0..149 of triangle, flattened

P. Bala, Notes on generalized Riordan arrays

P. Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), #13.5.1.

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Ik-Pyo Kim, 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

CROSSREFS

Rows 0-14 are A000984, A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.

Columns are A000217, A000292, A000332, A000389, A000579.

Diagonals are A005809, A045721, A025174, A004319, A013698, A003408.

Cf. A100100.

Sequence in context: A239102 A239103 A246971 * A128741 A175757 A060539

Adjacent sequences:  A092389 A092390 A092391 * A092393 A092394 A092395

KEYWORD

nonn,tabl,easy

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

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Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)