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A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

%I A185732 #14 Jul 12 2017 03:18:06
%S A185732 1,4,2,10,9,3,20,24,15,4,35,50,42,22,5,56,90,90,64,30,6,84,147,165,
%T A185732 140,90,39,7,120,224,273,260,200,120,49,8,165,324,420,434,375,270,154,
%U A185732 60,9,220,450,612,672,630,510,350,192,72,10,286,605,855,984,980,861,665,440,234,85,11,364,792,1155,1380,1440,1344,1127,840,540,280,99,12,455,1014,1518,1870,2025,1980,1764,1428,1035,650,330,114,13,560,1274,1950,2464,2750,2790,2604,2240
%N A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.
%C A185732 This is the (first) accumulation array of A086270; the second is A185733.  See A144112 for the definition of accumulation array.
%H A185732 G. C. Greubel, <a href="/A185732/b185732.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185732 T(n,k) = k*(k+1)*n*(n+1)*(k*n-n+k+5)/12.
%e A185732 Northwest corner:
%e A185732 1....4....10...20...35
%e A185732 2....9....24...50...90
%e A185732 3....15...42...90...165
%e A185732 4....22...64...140..260
%e A185732 5....30...90...200..375
%t A185732 f[n_,k_]:=k+n*k(k-1)/2;
%t A185732 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)
%t A185732 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)
%t A185732 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
%t A185732 Factor[s[n,k]]  (* formula for A185732 *)
%t A185732 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)
%t A185732 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)
%Y A185732 Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.
%Y A185732 Columns 1 to 4: A000027, A055999, A067728, 10*A000096.
%Y A185732 Cf. A144112, A086270, A185732.
%K A185732 nonn,tabl
%O A185732 1,2
%A A185732 _Clark Kimberling_, Feb 01 2011