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%I #85 Mar 13 2023 13:03:24
%S 1,0,1,0,1,1,0,0,2,1,0,0,1,3,1,0,0,0,3,4,1,0,0,0,1,6,5,1,0,0,0,0,4,10,
%T 6,1,0,0,0,0,1,10,15,7,1,0,0,0,0,0,5,20,21,8,1,0,0,0,0,0,1,15,35,28,9,
%U 1,0,0,0,0,0,0,6,35,56,36,10,1,0,0,0,0,0,0,1,21,70,84,45,11,1,0,0,0,0
%N Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
%C The signed triangular matrix T(n,k)*(-1)^(n-k) is the inverse matrix of the triangular Catalan convolution matrix A106566(n,k), n=k>=0, with A106566(n,k) = 0 if n<k. - _Philippe Deléham_, Aug 01 2005
%C As a number triangle: unsigned version of A109466. - _Philippe Deléham_, Oct 26 2008
%C A063967*A130595 as infinite lower triangular matrices. - _Philippe Deléham_, Dec 11 2008
%C Modulo 2, this sequence becomes A106344. - _Philippe Deléham_, Dec 18 2008
%C Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n) = Sum_{i=0..k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A111808. For example, s_1(n) = binomial(n,1) = n is the first column of A111808 for n>1, s_2(n) = binomial(n,1) + binomial(n,2) is the second column of A111808 for n>1, etc. Therefore, in cases k=3,4,5,6,7,8, s_k(n) is A005581(n), A005712(n), A000574(n), A005714(n), A005715(n), A005716(n), respectively. Besides, s_k(n+5) = A064054(n). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012
%C As a triangle, T(n,k) = binomial(k,n-k). - _Peter Bala_, Nov 27 2015
%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 Muniru A Asiru, <a href="/A026729/b026729.txt">Rows n=0..50 of triangle, flattened</a>
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</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 Lili Mu and Sai-nan Zheng, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Zheng/zheng8.html">On the Total Positivity of Delannoy-Like Triangles</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.
%H L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan Group</a>, Discrete Appl. Maths. 34 (1991) 229-239.
%F As a number triangle, this is defined by T(n,0) = 0^n, T(0,k) = 0^k, T(n,k) = T(n-1,k-1) + Sum_{j, j>=0} = (-1)^j*T(n-1,k+j)*A000108(j) for n>0 and k>0. - _Philippe Deléham_, Nov 07 2005
%F As a triangle read by rows, it is [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 22 2006
%F As a number triangle, this is defined by T(n, k) = Sum_{i=0..n} (-1)^(n+i)binomial(n, i)binomial(i+k, i-k) and is the Riordan array ( 1, x*(1+x) ). The row sums of this triangle are F(n+1). - _Paul Barry_, Jun 21 2004
%F Sum_{k=0..n}x^k*T(n,k) = A000007(n), A000045(n+1), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for n=0,1,2,3,4,5,6,7,8,9,10. - _Philippe Deléham_, Oct 16 2006
%F T(n,k) = A109466(n,k)*(-1)^(n-k). - _Philippe Deléham_, Dec 11 2008
%F G.f. for the triangular interpretation: -1/(-1+x*y+x^2*y). - _R. J. Mathar_, Aug 11 2015
%F For T(0,0) = 0, the triangle below has the o.g.f. G(x,t) = [t*x(1+x)]/[1-t*x(1+x)]. See A109466 for a signed version and inverse, A030528 for reverse and A102426 for a shifted version. - _Tom Copeland_, Jan 19 2016
%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 As a triangle, this begins
%e 1
%e 0 1
%e 0 1 1
%e 0 0 2 1
%e 0 0 1 3 1
%e 0 0 0 3 4 1
%e 0 0 0 1 6 5 1
%e ...
%e Production array is
%e 0 1
%e 0 1 1
%e 0 -1 1 1
%e 0 2 -1 1 1
%e 0 -5 2 -1 1 1
%e 0 14 -5 2 -1 1 1
%e 0 -42 14 -5 2 -1 1 1
%e 0 132 -42 14 -5 2 -1 1 1
%e 0 -429 132 -42 14 -5 2 -1 1 1
%e ... (Cf. A000108)
%p seq(seq(binomial(k,n-k),k=0..n),n=0..12); # _Peter Luschny_, May 31 2014
%t Table[Binomial[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Nov 28 2015 *)
%o (Magma) /* As triangle: */ [[Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Nov 29 2015
%o (GAP) nmax:=15;; T:=List([0..nmax],n->List([0..nmax],k->Binomial(n,k)));;
%o b:=List([2..nmax],n->OrderedPartitions(n,2));;
%o a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][1]][b[i][j][2]]))); # _Muniru A Asiru_, Jul 17 2018
%Y The official entry for Pascal's triangle is A007318. See also A052553 (the same array read by upward antidiagonals).
%Y Cf. A030528 (subtriangle for 1<=k<=n).
%Y Cf. A109466, A102426.
%K nonn,tabl,easy
%O 0,9
%A _N. J. A. Sloane_, Jan 19 2003
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