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%I #51 Sep 22 2023 08:04:59
%S 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,
%T 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,
%U 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
%N Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
%C Another version of Pascal's triangle A007318.
%C 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
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A052553/b052553.txt">Table of n, a(n) for n = 0..5459</a>
%H Math Stack Exchange, <a href="https://math.stackexchange.com/questions/1779632/homology-of-the-n-torus-using-the-künneth-formula">Homology of the n-torus using the Künneth Formula</a>
%H Nicholas Owad and Anastasiia Tsvietkova, <a href="https://arxiv.org/abs/2205.03451">Random meander model for links</a>, arXiv:2205.03451 [math.GT], 2022.
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F As a triangle: T(n,k) = A026729(n,n-k).
%F G.f. of the triangular version: 1/(1-x-x^2*y). - _R. J. Mathar_, Aug 11 2015
%e Array begins:
%e 1, 0, 0, 0, 0, 0, ...
%e 1, 1, 0, 0, 0, 0, ...
%e 1, 2, 1, 0, 0, 0, ...
%e 1, 3, 3, 1, 0, 0, ...
%e 1, 4, 6, 4, 1, 0, ...
%e 1, 5, 10, 10, 5, 1, ...
%e As a triangle, this begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 2, 0, 0;
%e 1, 3, 1, 0, 0;
%e 1, 4, 3, 0, 0, 0;
%e 1, 5, 6, 1, 0, 0, 0;
%e 1, 6, 10, 4, 0, 0, 0, 0;
%e ...
%p 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
%t Flatten[ Table[ Binomial[n-k , k], {n, 0, 13}, {k, 0, n}]] (* _Jean-François Alcover_, Dec 05 2012 *)
%o (PARI) T(n,k) = binomial(n,k) \\ _Charles R Greathouse IV_, Feb 07 2017
%o (Magma) /* As triangle */ [[Binomial(n-k,k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Feb 08 2017
%Y The official entry for Pascal's triangle is A007318. See also A026729 (the same array read by downward antidiagonals).
%Y As triangle without zeroes: A011973.
%Y Cf. A052509, A054123, A054124, A008949.
%K nonn,tabl,easy,nice
%O 0,8
%A _N. J. A. Sloane_, Mar 17 2000
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