A123110 Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Author

Philippe Deléham, Sep 28 2006

Keywords

• nonn
• tabl

Comments

Diagonal sums give A123108. - Philippe Deléham, Oct 08 2009

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle
• Index entries for characteristic functions.

Crossrefs

Essentially the same sequence as A114607.

Also essentially the same as A023532. - R. J. Mathar, Jun 18 2008

After the initial a(0)=1, the characteristic function of A014132.

Cf. A010054.

Programs

PARI

A123110(n) = (!n || !ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

Formula

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A028310(n), A095121(n), A123109(n) for x=0,1,2,3 respectively.

G.f.: (1-x+y*x^2)/(1-(1+y)*x+y*x^2). - Philippe Deléham, Nov 01 2011

From Tom Copeland, Nov 10 2012: (Start)

O.g.f. for row polynomials: 1 + (t/(1-t))*(1/(1-x)-1/(1-x*t)) = 1 + t*x + (t+t^2)*x^2 + ....

E.g.f. for row polynomials: 1 + (t/(1-t))*(e^x-e^(t*x)) = 1 + t*x + (t+t^2)*x^2/2 + .... (End)

a(0) = 1; for n > 0, a(n) = 1 - A010054(n). [As a flat sequence] - Antti Karttunen, Jan 19 2025