A143844 - OEIS

%I #16 Feb 21 2024 18:00:48

%S 0,0,1,0,1,4,0,1,4,9,0,1,4,9,16,0,1,4,9,16,25,0,1,4,9,16,25,36,0,1,4,

%T 9,16,25,36,49,0,1,4,9,16,25,36,49,64,0,1,4,9,16,25,36,49,64,81,0,1,4,

%U 9,16,25,36,49,64,81,100,0,1,4,9,16,25,36,49,64,81,100,121

%N Triangle T(n,k) = k^2 read by rows.

%C This is triangle A133819 with an additional leading column of zeros.

%C There is a family of even integer-valued polynomials p_n(x) = product_{k=0..n} (x^2 - T(n,k))/ A002674(n+1). We find p_0(x) in A000290, p_1(x) in A002415, p_2(x) essentially in A040977, p_3(x) in A053347 and p_4(x) in A054334. - _Paul Curtz_, Jun 10 2011

%F T(n,k) = (A002262(n,k))^2.

%F G.f.: x*y*(1 + x*y)/((1 - x)*(1 - x*y)^3). - _Stefano Spezia_, Feb 21 2024

%t Table[Range[0,n]^2,{n,0,15}]//Flatten (* _Harvey P. Dale_, Sep 08 2017 *)

%o (PARI) for(n=0,9,for(k=0,n,print1(k^2", "))) \\ _Charles R Greathouse IV_, Jun 10 2011

%Y Cf. A000290, A002262, A002415, A002674, A040977, A053347, A054334, A133819.

%K nonn,tabl,easy

%O 0,6

%A _Paul Curtz_, Sep 03 2008

%E Definition simplified by _R. J. Mathar_, Sep 07 2009