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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 February 24 14:09 EST 2024. Contains 370303 sequences. (Running on oeis4.)