A123110 - OEIS

%I #30 Jan 19 2025 20:46:43

%S 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,

%T 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,

%U 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

%N 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.

%C Diagonal sums give A123108. - _Philippe Deléham_, Oct 08 2009

%H Antti Karttunen, <a href="/A123110/b123110.txt">Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.

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

%F G.f.: (1-x+y*x^2)/(1-(1+y)*x+y*x^2). - _Philippe Deléham_, Nov 01 2011

%F From _Tom Copeland_, Nov 10 2012: (Start)

%F 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 + ....

%F 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)

%F a(0) = 1; for n > 0, a(n) = 1 - A010054(n). [As a flat sequence]  - _Antti Karttunen_, Jan 19 2025

%e Triangle begins:

%e   1;

%e   0, 1;

%e   0, 1, 1;

%e   0, 1, 1, 1;

%e   0, 1, 1, 1, 1;

%e   0, 1, 1, 1, 1, 1;

%e   0, 1, 1, 1, 1, 1, 1;

%e   0, 1, 1, 1, 1, 1, 1, 1;

%e   0, 1, 1, 1, 1, 1, 1, 1, 1;

%e   0, 1, 1, 1, 1, 1, 1, 1, 1, 1;

%o (PARI) A123110(n) = (!n || !ispolygonal(n,3)); \\ _Antti Karttunen_, Jan 19 2025

%Y Essentially the same sequence as A114607.

%Y Also essentially the same as A023532. - _R. J. Mathar_, Jun 18 2008

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

%Y Cf. A010054.

%K nonn,tabl

%O 0,1

%A _Philippe Deléham_, Sep 28 2006