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 A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by antidiagonals. 36
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS 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=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 As a triangle, T(n,k) = binomial(k,n-k). - Peter Bala, Nov 27 2015 LINKS Muniru A Asiru, Rows n=0..50 of triangle, flattened T. Copeland, Addendum to Elliptic Lie Triad Lili Mu and Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6. L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239. FORMULA 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 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 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 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 T(n,k) = A109466(n,k)*(-1)^(n-k). - Philippe Deléham, Dec 11 2008 G.f. for the triangular interpretation: -1/(-1+x*y+x^2*y). - R. J. Mathar, Aug 11 2015 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 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 ... As a triangle, this begins 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 ... Production array is 0    1 0    1   1 0   -1   1   1 0    2  -1   1  1 0   -5   2  -1  1  1 0   14  -5   2 -1  1  1 0  -42  14  -5  2 -1  1  1 0  132 -42  14 -5  2 -1  1  1 0 -429 132 -42 14 -5  2 -1  1  1 ... (Cf. A000108) MAPLE seq(seq(binomial(k, n-k), k=0..n), n=0..12); # Peter Luschny, May 31 2014 MATHEMATICA Table[Binomial[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 28 2015 *) PROG (MAGMA) /* As triangle: */ [[Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 29 2015 (GAP) nmax:=15;; T:=List([0..nmax], n->List([0..nmax], k->Binomial(n, k)));; b:=List([2..nmax], n->OrderedPartitions(n, 2));; 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 CROSSREFS The official entry for Pascal's triangle is A007318. See also A052553. Cf. A030528 (subtriangle for 1<=k<=n). Cf. A109466, A102426. Sequence in context: A319572 A108063 A164846 * A109466 A259095 A326676 Adjacent sequences:  A026726 A026727 A026728 * A026730 A026731 A026732 KEYWORD nonn,tabl,easy AUTHOR N. J. A. Sloane, Jan 19 2003 STATUS approved

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)