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A026729
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Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by antidiagonals.
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24
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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; internal format)
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OFFSET
| 0,9
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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 DELEHAM Aug 01 2005
As a number triangle : unsigned version of A109466 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 26 2008]
A063967*A130595 as infinite lower triangular matrices . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 11 2008]
Modulo 2, this sequence becomes A106344 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 18 2008]
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REFERENCES
| L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
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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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2006
As a number triangle, this is defined by T(n, k)=sum{i=0..n, (-1)^(n+i)C(n, i)C(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 (pbarry(AT)wit.ie), Jun 21 2004
Sum_{k, 0<=k<=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 DELEHAM (kolotoko(AT)wanadoo.fr), Oct 16 2006
T(n,k)= A109466(n,k)*(-1)^(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 11 2008]
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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
...
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CROSSREFS
| The official entry for Pascal's triangle is A007318. See also A052553.
Cf. A030528 (subtriangle for 1<=k<=n).
Sequence in context: A108063 A164846 * A109466 A076833 A071676 A115363
Adjacent sequences: A026726 A026727 A026728 * A026730 A026731 A026732
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KEYWORD
| nonn,tabl,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 19 2003
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EXTENSIONS
| More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 19 2003
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