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A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by antidiagonals. 11
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 0, 1, 12, 55 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Another version of Pascal's triangle A007318.

As a triangle read by rows, it is (1,0,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938 and it is the Riordan array (1/(1-x), x^2/(1-x)). The row sums of this triangle are F(n+1) = A000045(n+1). - Philippe Deléham, Dec 11 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5459

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

As a triangle: T(n,k) = A026729(n,n-k).

G.f. of the triangular version: 1/(1-x-x^2*y). - R. J. Mathar, Aug 11 2015

EXAMPLE

Array begins:

1 0  0  0 0 0 ...

1 1  0  0 0 0 ...

1 2  1  0 0 0 ...

1 3  3  1 0 0 ...

1 4  6  4 1 0 ...

1 5 10 10 5 1 ...

As a triangle, this begins:

1

1, 0

1, 1,  0

1, 2,  0, 0

1, 3,  1, 0, 0

1, 4,  3, 0, 0, 0

1, 5,  6, 1, 0, 0, 0

1, 6, 10, 4, 0, 0, 0, 0

MAPLE

with(combinat): for s from 0 to 20 do for n from s to 0 by -1 do printf(`%d, `, binomial(n, s-n)) od:od: # James A. Sellers, Mar 17 2000

MATHEMATICA

Flatten[ Table[ Binomial[n-k , k], {n, 0, 13}, {k, 0, n}]]  (* Jean-François Alcover, Dec 05 2012 *)

PROG

(PARI) T(n, k) = binomial(n, k) \\ Charles R Greathouse IV, Feb 07 2017

(MAGMA) /* As triangle */ [[Binomial(n-k, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 08 2017

CROSSREFS

The official entry for Pascal's triangle is A007318. See also A026729.

Cf. A052509, A054123, A054124, A008949.

Sequence in context: A216599 A114510 A077029 * A290054 A290430 A290429

Adjacent sequences:  A052550 A052551 A052552 * A052554 A052555 A052556

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane, Mar 17 2000

STATUS

approved

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Last modified February 21 12:27 EST 2018. Contains 299411 sequences. (Running on oeis4.)