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A185872 Accumulation array of the (odd,odd)-polka dot array A185868, by antidiagonals.

%I A185872 #12 Jul 17 2017 02:31:54
%S A185872 1,5,7,16,24,22,38,59,65,50,75,120,141,136,95,131,215,262,274,245,161,
%T A185872 210,352,440,480,470,400,252,316,539,687,770,790,741,609,372,453,784,
%U A185872 1015,1160,1225,1208,1099,880,525,625,1095,1436,1666,1795,1825,1750,1556,1221,715,836,1480,1962,2304,2520,2616,2590,2432,2124,1640,946,1090,1947,2605,3090,3420,3605,3647,3540
%N A185872 Accumulation array of the (odd,odd)-polka dot array A185868, by antidiagonals.
%C A185872 See A144112 for the definition of accumulation array.
%H A185872 G. C. Greubel, <a href="/A185872/b185872.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185872 T(n,k) = (k*n/6)*(4*n^2 + 6*n*k + 4*k^2 - 3*n - 9*k + 4), k>=1, n>=1.
%e A185872 Northwest corner:
%e A185872    1,   5,  16,  38,  75
%e A185872    7,  24,  59, 120, 215
%e A185872   22,  54, 141, 262, 440
%e A185872   50, 136, 174, 480, 770
%t A185872 f[n_,k_]:=2n-1+(2n+2k-4)(2n+2k-3)/2;
%t A185872 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185868 *)
%t A185872 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185872 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185872 FullSimplify[s[n,k]] (*formula for A185872 *)
%t A185872 g[n_]:=Sum[f[n+1-k,k],{k,1,n}];
%t A185872 Table[g[n],{n,50}] (* A185872 *)
%t A185872 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
%Y A185872 Cf. A185868.
%Y A185872 Row 1: A174723; column 1: A002412.
%K A185872 nonn,tabl
%O A185872 1,2
%A A185872 _Clark Kimberling_, Feb 05 2011