A125091 Triangle read by rows: T(n,k) = (1/6)*k*(k+1)*(k+2)*binomial(n,k) (1 <= k <= n).

1, 2, 4, 3, 12, 10, 4, 24, 40, 20, 5, 40, 100, 100, 35, 6, 60, 200, 300, 210, 56, 7, 84, 350, 700, 735, 392, 84, 8, 112, 560, 1400, 1960, 1568, 672, 120, 9, 144, 840, 2520, 4410, 4704, 3024, 1080, 165, 10, 180, 1200, 4200, 8820, 11760, 10080, 5400, 1650, 220, 11

Offset: 1

Author

Gary W. Adamson, Nov 19 2006

Keywords

• nonn
• tabl

Comments

T(n,n) = n*(n+1)*(n+2)/6 = A000292(n).

Sum_{k=1..n} T(n,k) = 2^n*n*(n+2)*(n+7)/48 = A055585(n-1).

Examples

			Triangle starts:
  1;
  2,   4;
  3,  12,  10;
  4,  24,  40,  20;
  5,  40, 100, 100,  35;
  6,  60, 200, 300, 210,  56;
  7,  84, 350, 700, 735, 392,  84;
		

Crossrefs

Cf. A055585.

Cf. A000292.

Programs

Maple

T:=(n,k)->k*(k+1)*(k+2)*binomial(n,k)/6: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[(k(k+1)(k+2)Binomial[n,k])/6,{n,20},{k,n}]] (* Harvey P. Dale, Jan 23 2016 *)

Extensions

Edited by N. J. A. Sloane, Dec 04 2006