A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

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

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

As a triangle, binomial(n-k, k) is also the number of ways to add k pierced circles to a path graph P_n so that no two circles share a vertex (see Lemma 3.1 at page 5 in Owad and Tsvietkova). - Stefano Spezia, May 18 2022

For all n >= 0, k >= 0, the k-th homology group of the n-torus H_k(T^n) is the free abelian group of rank T(n,k) = binomial(n,k). See the Math Stack Exchange link below. - Jianing Song, Mar 13 2023

Links

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

Math Stack Exchange, Homology of the n-torus using the Künneth Formula

Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.

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 (the same array read by downward antidiagonals).

As triangle without zeroes: A011973.

Cf. A052509, A054123, A054124, A008949.

Sequence in context: A114510 A325466 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