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A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by antidiagonals. 12

%I

%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 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

%H Vincenzo Librandi, <a href="/A052553/b052553.txt">Table of n, a(n) for n = 0..5459</a>

%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

%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.

%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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Last modified January 27 04:57 EST 2020. Contains 331291 sequences. (Running on oeis4.)