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A103451 Triangular array T read by rows: T(n, 0) = T(n, n) = 1, T(n, k) = 0 for 0 <= k <= n. 32
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Equals Pascal's triangle (A007318) where all elements > 1 are replaced with zero. Therefore it might be called "binomial skeleton".

Row sums are in A040000, antidiagonal sums are in A040001. When construed as a lower triangular matrix, the matrix inverse is A103452.

LINKS

Michael De Vlieger, Rows n = 0..140 of triangle, flattened

Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731.

FORMULA

a(n) = A097806(n-1) for n > 0. - Philippe Deléham, Oct 16 2007

T(n,k) = C(n,k-n) + C(n,-k) - C(0,n+k), 0 <= k <= n. - Eric Werley, Jul 01 2011

EXAMPLE

First few rows are:

  1;

  1, 1;

  1, 0, 1;

  1, 0, 0, 1;

  1, 0, 0, 0, 1;

  1, 0, 0, 0, 0, 1;

  ...

MATHEMATICA

Table[Boole[n == 0 || Mod[k, n] == 0], {n, 0, 14}, {k, 0, n}] (* or *)

Table[Binomial[n, k - n] + Binomial[n, -k] - Binomial[0, n + k], {n, 0, 14}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

PROG

(MAGMA) r:=14; T:=ScalarMatrix(r, 1); for n in [1..r] do T[n, 1]:=1; end for; &cat[ [ T[n, k]: k in [1..n] ]: n in [1..r] ];

(MAGMA) /* As triangle */ [[Binomial(n, k-n)+Binomial(n, -k)-Binomial(0, n+k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 20 2016

(PARI) for(n=0, 15, for(k=0, n, print1(binomial(n, k-n) + binomial(n, -k) - binomial(0, n+k), ", "))) \\ G. C. Greubel, Dec 08 2018

CROSSREFS

Cf. A007318, A040000, A040001, A103452, A097806.

Sequence in context: A189289 A270885 A127972 * A103452 A131219 A127970

Adjacent sequences:  A103448 A103449 A103450 * A103452 A103453 A103454

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Feb 06 2005

EXTENSIONS

Edited by Klaus Brockhaus, Jan 26 2011

STATUS

approved

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Last modified October 19 12:21 EDT 2019. Contains 328220 sequences. (Running on oeis4.)