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

1, 4, 2, 10, 9, 3, 20, 24, 15, 4, 35, 50, 42, 22, 5, 56, 90, 90, 64, 30, 6, 84, 147, 165, 140, 90, 39, 7, 120, 224, 273, 260, 200, 120, 49, 8, 165, 324, 420, 434, 375, 270, 154, 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

Offset: 1

Author

Clark Kimberling, Feb 01 2011

Keywords

• nonn
• tabl

Comments

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

Examples

			Northwest corner:
1....4....10...20...35
2....9....24...50...90
3....15...42...90...165
4....22...64...140..260
5....30...90...200..375
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

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

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

Cf. A144112, A086270, A185732.

Programs

Mathematica

f[n_,k_]:=k+n*k(k-1)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185732 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)

Formula

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