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.
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
KEYWORD
AUTHOR
N. J. A. Sloane, Mar 17 2000
STATUS
approved