

A052553


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


12



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, 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, 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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,8


COMMENTS

Another version of Pascal's triangle A007318.
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/(1x), x^2/(1x)). The row sums of this triangle are F(n+1) = A000045(n+1).  Philippe Deléham, Dec 11 2011


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5459
Index entries for triangles and arrays related to Pascal's triangle


FORMULA

As a triangle: T(n,k) = A026729(n,nk).
G.f. of the triangular version: 1/(1xx^2*y).  R. J. Mathar, Aug 11 2015


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 ...
1 5 10 10 5 1 ...
As a triangle, this begins:
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, 0, 0


MAPLE

with(combinat): for s from 0 to 20 do for n from s to 0 by 1 do printf(`%d, `, binomial(n, sn)) od:od: # James A. Sellers, Mar 17 2000


MATHEMATICA

Flatten[ Table[ Binomial[nk , k], {n, 0, 13}, {k, 0, n}]] (* JeanFrançois Alcover, Dec 05 2012 *)


PROG

(PARI) T(n, k) = binomial(n, k) \\ Charles R Greathouse IV, Feb 07 2017
(MAGMA) /* As triangle */ [[Binomial(nk, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 08 2017


CROSSREFS

The official entry for Pascal's triangle is A007318. See also A026729.
Cf. A052509, A054123, A054124, A008949.
Sequence in context: A114510 A325466 A077029 * A290054 A290430 A290429
Adjacent sequences: A052550 A052551 A052552 * A052554 A052555 A052556


KEYWORD

nonn,tabl,easy,nice


AUTHOR

N. J. A. Sloane, Mar 17 2000


STATUS

approved



