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

1,2

Comments

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

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

Links

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

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.

Example

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)

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

Crossrefs

Cf. A144112, A185914, A185916.

Sequence in context: A201904 A133110 A286158 * A086270 A325000 A104712

Adjacent sequences: A185912 A185913 A185914 * A185916 A185917 A185918

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 06 2011

Status

approved