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A068607 Triangle of T(n,k)=n*k*(n+k+1) with n>=k>=0. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)