A103219 Triangle read by rows: T(n,k) = (n+1-k)*(4*(n+1-k)^2 - 1)/3+2*k*(n+1-k)^2.

1, 10, 3, 35, 18, 5, 84, 53, 26, 7, 165, 116, 71, 34, 9, 286, 215, 148, 89, 42, 11, 455, 358, 265, 180, 107, 50, 13, 680, 553, 430, 315, 212, 125, 58, 15, 969, 808, 651, 502, 365, 244, 143, 66, 17, 1330, 1131, 936, 749, 574, 415, 276, 161, 74, 19, 1771, 1530, 1293

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 26 2005

Keywords

• nonn
• tabl
• easy

Comments

The triangle is generated from the product B * A of the infinite lower triangular matrices 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,
10,3,
35,18,5,
84,53,26,7,
165,116,71,34,9,
286,215,148,89,42,11,
		

Crossrefs

Row sums give A103220.

T(n, 0) = (n+1)*(4*(n+1)^2 - 1)/3 = A000447(n+1);

T(n+1, n)= 8*n+2 = A017089(n+1);

Cf. A103218 (for product A*B), A103220.

Programs

Mathematica

T[n_, k_] := (n + 1 - k)*(4*(n + 1 - k)^2 - 1)/3 + 2*k*(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)=(n+1-k)*(4*(n+1-k)^2-1)/3+2*k*(n+1-k)^2; for(i=0,10, for(j=0,i,print1(T(i,j),","));print())