A143844 Triangle T(n,k) = k^2 read by rows.

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

Offset: 0

Author

Paul Curtz, Sep 03 2008

Keywords

• nonn
• tabl
• easy

Comments

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

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

Crossrefs

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

Programs

Mathematica

Table[Range[0,n]^2,{n,0,15}]//Flatten (* Harvey P. Dale, Sep 08 2017 *)

PARI

for(n=0,9,for(k=0,n,print1(k^2", "))) \\ Charles R Greathouse IV, Jun 10 2011

Formula

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

G.f.: x*y*(1 + x*y)/((1 - x)*(1 - x*y)^3). - Stefano Spezia, Feb 21 2024

Extensions

Definition simplified by R. J. Mathar, Sep 07 2009