A103451 Triangular array T read by rows: T(n, 0) = T(n, n) = 1, T(n, k) = 0 for 0 < k < n.

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

Offset: 0

Author

Paul Barry, Feb 06 2005

Keywords

• easy
• nonn
• tabl

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.

Examples

			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;
  ...
		

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.
• Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
• Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 5.

Crossrefs

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

Programs

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

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 *)

PARI

for(n=0,15, for(k=0,n, print1(if(k==0||k==n, 1, 0), ", "))) \\ G. C. Greubel, Dec 08 2018

Python

from math import isqrt, comb
def A103451(n):
    if n==0: return 1
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return int(not (n-comb(a+1,2))%a) # Chai Wah Wu, Jun 24 2025

Sage

def A103451(n,k): return 1 if (k==0 or k==n) else 0
flatten([[A103451(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 14 2021

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

From Stefano Spezia, Jul 04 2024: (Start)

G.f.: (1 - x^2*y)/((1 - x)*(1 - x*y)).

E.g.f.: BesselI(0, 2*sqrt(x*y)) + exp(x) - 1. (End)

Extensions

Edited by Klaus Brockhaus, Jan 26 2011