A185734 Third accumulation array of the polygonal number array (A086270), by antidiagonals.

1, 6, 4, 21, 25, 10, 56, 90, 65, 20, 126, 245, 240, 135, 35, 252, 560, 665, 510, 245, 56, 462, 1134, 1540, 1435, 945, 406, 84, 792, 2100, 3150, 3360, 2695, 1596, 630, 120, 1287, 3630, 5880, 6930, 6370, 4606, 2520, 930, 165, 2002, 5940, 10230

Offset

1,2

Comments

The chain of accumulation arrays is A144257->A086270->A185732->A185733->A184734.

Links

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

Formula

T(n,k) = C(k+3,4)*C(n+2,3)*(k*n-n+3*k+17)/20, k>=1, n>=1.

T(n,k) = Sum_{j=1..n} Sum_{l=1..k} A185733(j,l), by definition.

Example

Northwest corner:
1....6......21.....56....126
4....25.....90....245....560
10...65....240....665...1540
20...135...510...1435...3360

Mathematica

f[n_, k_]:= k*(1+k)*(2+k)*n*(1+n)*(10+2*k-n+k*n)/144;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]  (* array A185733 *)
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten  (* A185733 *)
s[n_, k_]:=Sum[f[i, j], {i, 1, n}, {j, 1, k}]; (* accumulation array of {f(n, k)} *)
FullSimplify[s[n, k]]  (* the formula for A185734 *)
TableForm[Table[s[n, k], {n, 1, 10}, {k, 1, 15}]]  (* array A185734 *)
Table[s[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten  (* A185734 *)

Crossrefs

Cf. A144112, A144257, A086270, A185732, A185733.

Sequence in context: A009278 A213573 A321417 * A292696 A318209 A120462

Adjacent sequences: A185731 A185732 A185733 * A185735 A185736 A185737

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Feb 02 2011

Status

approved