A185915 Accumulation array of A185914, by antidiagonals.

1, 3, 1, 6, 4, 1, 10, 9, 4, 1, 15, 16, 10, 4, 1, 21, 25, 19, 10, 4, 1, 28, 36, 31, 20, 10, 4, 1, 36, 49, 46, 34, 20, 10, 4, 1, 45, 64, 64, 52, 35, 20, 10, 4, 1, 55, 81, 85, 74, 55, 35, 20, 10, 4, 1, 66, 100, 109, 100, 80, 56, 35, 20, 10, 4, 1, 78, 121, 136, 130, 110, 83, 56, 35, 20, 10, 4, 1, 91, 144, 166, 164, 145, 116, 84, 56, 35, 20, 10, 4, 1, 105, 169, 199, 202, 185, 155, 119, 84, 56, 35, 20, 10, 4, 1

Offset: 1

Author

Clark Kimberling, Feb 06 2011

Keywords

• nonn
• tabl

Comments

A member of the accumulation chain ... < A185916 < A185914 < A185915 < ...

(See A144112 for definitions of weight array and accumulation array.)

Examples

			Northwest corner:
1....3....6....10....15....21....28
1....4....9....16....25....36....49
1....4....10...19....31....46....64
1....4....10...20....34....52....74
1....4....10...20....35....55....80
1....4....10...20....35....56....83
row 1: A000217 (triangular numbers)
row 2: A000290 (squares)
row 3: A005448 (centered triangular numbers)
row 4: A005893
row 5: A062985
Limit of rows: A000292 (tetrahedral numbers)
		

Links

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

Crossrefs

Cf. A144112, A185914, A185916.

Programs

Mathematica

f[n_, 0] := 0; f[0, k_] := 0; f[n_, k_] := k - n + 1; f[n_, k_] := 0 /; k < n; s[n_, k_] := Sum[f[i, j], {i, 1, n}, {j, 1, k}]; Table[s[n - k + 1, k], {n, 50}, {k, n, 1, -1}] // Flatten

Formula

T(n,k) = C(k+2,3) if k<=n; T(n,k) = k*(k+2-n)/2 if k>n; k>=1, n>=1.