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

0,1

Comments

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

Links

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

Index entries for characteristic functions.

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

Example

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;

Prog

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

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.

Sequence in context: A213061 A110247 A114607 * A004593 A094934 A245837

Adjacent sequences: A123107 A123108 A123109 * A123111 A123112 A123113

Keyword

nonn,tabl

Author

Philippe Deléham, Sep 28 2006

Status

approved