A185732 - OEIS

%I #14 Jul 12 2017 03:18:06

%S 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 140,90,39,7,120,224,273,260,200,120,49,8,165,324,420,434,375,270,154,

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

%C This is the (first) accumulation array of A086270; the second is A185733.  See A144112 for the definition of accumulation array.

%H G. C. Greubel, <a href="/A185732/b185732.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n,k) = k*(k+1)*n*(n+1)*(k*n-n+k+5)/12.

%e Northwest corner:

%e 1....4....10...20...35

%e 2....9....24...50...90

%e 3....15...42...90...165

%e 4....22...64...140..260

%e 5....30...90...200..375

%t f[n_,k_]:=k+n*k(k-1)/2;

%t TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)

%t Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)

%t s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)

%t Factor[s[n,k]]  (* formula for A185732 *)

%t TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)

%t Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)

%Y Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.

%Y Columns 1 to 4: A000027, A055999, A067728, 10*A000096.

%Y Cf. A144112, A086270, A185732.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 01 2011