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

%I A068607 #8 Dec 08 2015 08:45:33
%S A068607 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 A068607 180,264,360,468,0,63,140,231,336,455,588,735,0,80,176,288,416,560,
%U A068607 720,896,1088,0,99,216,351,504,675,864,1071,1296,1539,0,120,260,420,600
%N A068607 Triangle of T(n,k)=n*k*(n+k+1) with n>=k>=0.
%C A068607 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 A068607 Harvey P. Dale, <a href="/A068607/b068607.txt">Table of n, a(n) for n = 0..1000</a>
%e A068607   0
%e A068607   0   3
%e A068607   0   8  20
%e A068607   0  15  36  63
%e A068607   0  24  56  96 144
%e A068607   0  35  80 135 200 275
%e A068607   0  48 108 180 264 360 468
%e A068607   0  63 140 231 336 455 588 735
%e A068607   0  80 176 288 416 560 720 896 1088
%t A068607 Flatten[Table[n*k*(n+k+1),{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, May 17 2015 *)
%Y A068607 Cf. A068606 for the same table as a square array.
%K A068607 nonn,tabl,easy
%O A068607 0,3
%A A068607 _Henry Bottomley_, Feb 24 2002