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

1,2

Comments

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

Links

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

Formula

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

Example

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

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 *)

Crossrefs

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

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

Cf. A144112, A086270, A185732.

Sequence in context: A066579 A117821 A266851 * A121794 A250108 A210735

Adjacent sequences: A185729 A185730 A185731 * A185733 A185734 A185735

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 01 2011

Status

approved