A103218 Triangle read by rows: T(n, k) = (2*k+1)*(n+1-k)^2.

1, 4, 3, 9, 12, 5, 16, 27, 20, 7, 25, 48, 45, 28, 9, 36, 75, 80, 63, 36, 11, 49, 108, 125, 112, 81, 44, 13, 64, 147, 180, 175, 144, 99, 52, 15, 81, 192, 245, 252, 225, 176, 117, 60, 17, 100, 243, 320, 343, 324, 275, 208, 135, 68, 19, 121, 300, 405, 448, 441, 396, 325, 240

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

Keywords

• nonn
• tabl

Comments

The triangle is generated from the product A * B of the infinite lower triangular matrix A =

1 0 0 0...

3 1 0 0...

5 3 1 0...

7 5 3 1...

... and B =

1 0 0 0...

1 3 0 0...

1 3 5 0...

1 3 5 7...

...

Examples

			Triangle begins:
1,
4,3,
9,12,5,
16,27,20,7,
25,48,45,28,9,
		

Crossrefs

Row sums give A002412 (hexagonal pyramidal numbers).

T(n, 0)=A000290(n+1) (the squares);

T(n, 1)=3*n^2=A033428(n);

T(n, 2)=5*n^2=A033429(n+1);

T(n, 3)=7*n^2=A033582(n+2);

Cf. A103219 (product B*A), A002412, A000290.

Programs

Mathematica

T[n_, k_] := (2*k + 1)*(n + 1 - k)^2; Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Feb 10 2005 *)

PARI

T(n, k) = (2*k+1)*(n+1-k)^2; for(i=0,10, for(j=0,i,print1(T(i,j),","));print())