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

Author

Clark Kimberling, Feb 02 2011

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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.