A068607 Triangle of T(n,k)=n*k*(n+k+1) with n>=k>=0.

0, 0, 3, 0, 8, 20, 0, 15, 36, 63, 0, 24, 56, 96, 144, 0, 35, 80, 135, 200, 275, 0, 48, 108, 180, 264, 360, 468, 0, 63, 140, 231, 336, 455, 588, 735, 0, 80, 176, 288, 416, 560, 720, 896, 1088, 0, 99, 216, 351, 504, 675, 864, 1071, 1296, 1539, 0, 120, 260, 420, 600

Offset

0,3

Comments

Considering partitions with up to n positive integers each no more than k (or equivalently paths of length n+k from one corner to the opposite corner of an n*k rectangle) there are C(n+k,n) such partitions (or paths); the mean of the sums of the partitions (or mean of the areas above the paths) is nk/2; and the variance of the sums of the partitions (or variance of the areas above the paths) is a(n)/12.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Example

  0
  0   3
  0   8  20
  0  15  36  63
  0  24  56  96 144
  0  35  80 135 200 275
  0  48 108 180 264 360 468
  0  63 140 231 336 455 588 735
  0  80 176 288 416 560 720 896 1088

Mathematica

Flatten[Table[n*k*(n+k+1), {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, May 17 2015 *)

Crossrefs

Cf. A068606 for the same table as a square array.

Sequence in context: A095123 A019691 A192919 * A303633 A167004 A287632

Adjacent sequences: A068604 A068605 A068606 * A068608 A068609 A068610

Keyword

nonn,tabl,easy

Author

Henry Bottomley, Feb 24 2002

Status

approved