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A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by antidiagonals. 35
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<k. - Philippe Deléham, Aug 01 2005

As a number triangle: unsigned version of A109466. - Philippe Deléham, Oct 26 2008

A063967*A130595 as infinite lower triangular matrices. - Philippe Deléham, Dec 11 2008

Modulo 2, this sequence becomes A106344. - Philippe Deléham, Dec 18 2008

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

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: A108063 A164846 * A109466 A259095 A076833 A071676

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 September 20 00:54 EDT 2018. Contains 315220 sequences. (Running on oeis4.)