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A068607 Triangle of T(n,k)=n*k*(n+k+1) with n>=k>=0. 3

%I #8 Dec 08 2015 08:45:33

%S 0,0,3,0,8,20,0,15,36,63,0,24,56,96,144,0,35,80,135,200,275,0,48,108,

%T 180,264,360,468,0,63,140,231,336,455,588,735,0,80,176,288,416,560,

%U 720,896,1088,0,99,216,351,504,675,864,1071,1296,1539,0,120,260,420,600

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

%C 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.

%H Harvey P. Dale, <a href="/A068607/b068607.txt">Table of n, a(n) for n = 0..1000</a>

%e 0

%e 0 3

%e 0 8 20

%e 0 15 36 63

%e 0 24 56 96 144

%e 0 35 80 135 200 275

%e 0 48 108 180 264 360 468

%e 0 63 140 231 336 455 588 735

%e 0 80 176 288 416 560 720 896 1088

%t Flatten[Table[n*k*(n+k+1),{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, May 17 2015 *)

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

%K nonn,tabl,easy

%O 0,3

%A _Henry Bottomley_, Feb 24 2002

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Last modified April 15 00:51 EDT 2024. Contains 371667 sequences. (Running on oeis4.)