A185733 -id:A185733 - cp's OEIS frontend

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

Clark Kimberling, Feb 01 2011

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

			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

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

Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.
Columns 1 to 4: A000027, A055999, A067728, 10*A000096.
Cf. A144112, A086270, A185732.

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 *)

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

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

Original entry on oeis.org

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

Clark Kimberling, Feb 02 2011

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

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

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

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

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 *)

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.

cp's OEIS Frontend

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

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