A185780 -id:A185780 - cp's OEIS frontend

A185781 Accumulation array of A185780, by antidiagonals.

1, 5, 2, 14, 12, 3, 30, 36, 21, 4, 55, 80, 66, 32, 5, 91, 150, 150, 104, 45, 6, 140, 252, 285, 240, 150, 60, 7, 204, 392, 483, 460, 350, 204, 77, 8, 285, 576, 756, 784, 675, 480, 266, 96, 9, 385, 810, 1116, 1232, 1155, 930, 630, 336, 117, 10, 506, 1100, 1575, 1824, 1820, 1596, 1225, 800, 414, 140, 11, 650, 1452, 2145, 2580, 2700, 2520, 2107, 1560, 990, 500, 165, 12

Offset: 1

Clark Kimberling, Feb 03 2011

See A144112 and A185780.

			Northwest corner:
1.....5....14....30....55
2.....12...36....80....150
3.....21...66....150...285
4.....32...104...240...460
5.....45...150...350...675

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

Cf. A144112, A185780.

Columns 1 to 4: A000027, A028347, 2*A033537, 10*A005563.

(See A185780.)
f[n_, k_] := k*(k + 1)*n*(k*n - n + k + 2)/6; Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = k*(k+1)*n*(k*n-n+k+2)/6, k>=1, n>=1.

A185783 Second accumulation array of A185780, by antidiagonals.

Original entry on oeis.org

1, 6, 3, 20, 20, 6, 50, 70, 44, 10, 105, 180, 160, 80, 15, 196, 385, 420, 300, 130, 21, 336, 728, 910, 800, 500, 196, 28, 540, 1260, 1736, 1750, 1350, 770, 280, 36, 825, 2040, 3024, 3360, 2975, 2100, 1120, 384, 45, 1210, 3135, 4920, 5880, 5740, 4655, 3080, 1560, 510, 55, 1716, 4620, 7590, 9600, 10080, 9016, 6860, 4320, 2100, 660, 66, 2366, 6578, 11220, 14850, 16500, 15876, 13328, 9660, 5850, 2750, 836, 78

Offset: 1

Clark Kimberling, Feb 03 2011

See A144112 and A185780.

			Northwest corner:
1....6....20....50....105
3....20...70....180...385
6....44...160...420...910
10...80...300...800...1750

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

Cf. A144112, A185780.
Row 1: A002415 (4-dimensional pyramidal numbers).
Columns 1 to 3: A000217, 2*A006503, 10*A005581.

(See A185780.)
f[n_, k_] := Binomial[k + 2, 3]*Binomial[n + 1, 2]*(k*n - n + 2*k + 4)/6; Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = C(k+2,3)*C(n+1,2)*(k*n-n+2*k+4)/6, k>=1, n>=1.

A185782 Weight array of A185780, by antidiagonals.

Original entry on oeis.org

1, 3, 0, 5, 2, 0, 7, 4, 2, 0, 9, 6, 4, 2, 0, 11, 8, 6, 4, 2, 0, 13, 10, 8, 6, 4, 2, 0, 15, 12, 10, 8, 6, 4, 2, 0, 17, 14, 12, 10, 8, 6, 4, 2, 0, 19, 16, 14, 12, 10, 8, 6, 4, 2, 0, 21, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0, 23, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0, 25, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0, 27, 24, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0

Offset: 1

Clark Kimberling, Feb 03 2011

			Northwest corner:
1....3....5....7....9....11....13
0....2....4....6....8....10....12
0....2....4....6....8....10....12
(row n)=(row n-1) for n>=3.

Cf. A144112, A185780.

Mathematica
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(See A185780.)
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A185781 Accumulation array of A185780, by antidiagonals.

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A185783 Second accumulation array of A185780, by antidiagonals.

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A185782 Weight array of A185780, by antidiagonals.

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