A185910 -id:A185910 - cp's OEIS frontend

A185911 Weight array of A185910, by antidiagonals.

1, 1, 3, 1, 0, 5, 1, 0, 0, 7, 1, 0, 0, 0, 9, 1, 0, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 13, 1, 0, 0, 0, 0, 0, 0, 15, 1, 0, 0, 0, 0, 0, 0, 0, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...

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

			Northwest corner:
  1, 1, 1, 1, 1, 1
  3, 0, 0, 0, 0, 0
  5, 0, 0, 0, 0, 0
  7, 0, 0, 0, 0, 0
  9, 0, 0, 0, 0, 0

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

Cf. A144112, A185910, A185912.

f[n_, 0] := 0; f[0, k_] := 0; f[n_, k_] := n^2 + k - 1;
w[m_, n_] := f[m, n] + f[m - 1, n - 1] - f[m, n - 1] - f[m - 1, n] /; Or[m > 0, n > 0];
Table[w[n - k + 1, k], {n, 50}, {k, n, 1, -1}] // Flatten
T[1, k_] := 1; T[n_, 1] := 2*n - 1; T[n_, k_] := 0; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}]//Flatten (* G. C. Greubel, Jul 22 2017 *)

T(1,k) = 1 for k >= 1; T(n,1) = 2*n-1 for n >= 1; T(n,k) = 0 otherwise.

A185912 Accumulation array of A185910; by antidiagonals.

Original entry on oeis.org

1, 3, 5, 6, 12, 14, 10, 21, 31, 30, 15, 32, 51, 64, 55, 21, 45, 74, 102, 115, 91, 28, 60, 100, 144, 180, 188, 140, 36, 77, 129, 190, 250, 291, 287, 204, 45, 96, 161, 240, 325, 400, 441, 416, 285, 55, 117, 196, 294, 405, 515, 602, 636, 579, 385, 66, 140, 234, 352, 490, 636, 770, 864, 882, 780, 506, 78, 165, 275, 414, 580, 763, 945, 1100, 1194, 1185, 1023, 650, 91, 192, 319, 480, 675, 896, 1127, 1344, 1515, 1600, 1551, 1312, 819, 105

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...

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

			Northwest corner:
   1,   3,   6,  10,  15
   5,  12,  21,  32,  45
  14,  31,  51,  74, 100
  30,  64, 102, 144, 190

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

Cf. A144112, A185910, A185913.
Row 1 to 2: A000217, A028347.
Column 1 to 3: A000330, A037237, 3*A145066.

f[n_, 0] := 0; f[0, k_] := 0;
f[n_, k_] := n^2 + k - 1;
s[n_, k_] := Sum[f[i, j], {i, 1, n}, {j, 1, k}];(*accumulation array of {f(n,k)}*)
FullSimplify[s[n, k]]  (*formula for A185812*)
Table[s[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten
T[n_, k_] := (k*n/6)*(2*n^2 + 3*n + 3*k - 2) ; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

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

A185913 Accumulation array of A185912, by antidiagonals.

Original entry on oeis.org

1, 4, 6, 10, 21, 20, 20, 48, 66, 50, 35, 90, 144, 160, 105, 56, 150, 260, 340, 330, 196, 84, 231, 420, 600, 690, 609, 336, 120, 336, 630, 950, 1200, 1260, 1036, 540, 165, 468, 896, 1400, 1875, 2170, 2128, 1656, 825, 220, 630, 1224, 1960, 2730, 3360, 3640, 3384, 2520, 1210, 286, 825, 1620, 2640, 3780, 4851, 5600, 5760, 5130, 3685, 1716, 364, 1056, 2090, 3450, 5040, 6664, 8036, 8820, 8700, 7480, 5214, 2366, 455

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...

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

			Northwest corner:
1.....4.....10.....20.....35
6.....21....48.....90.....150
20....66....144....260....420
50....160...340....600....950

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

Cf. A144112, A185912.
Row 1 to 2: A000292, 3*A005581.
Column 1: A002415.

(* The program generates A185912 and its accumulation array A185913 *)
f[n_,k_]:=(k*n/6)(-2+3k+3n+2n^2);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* array A185912 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
FullSimplify[s[n,k]]  (* formula for A185913 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* array A185913 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

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

A055630 Table T(k,m) = k^2 + m read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 4, 2, 2, 9, 5, 3, 3, 16, 10, 6, 4, 4, 25, 17, 11, 7, 5, 5, 36, 26, 18, 12, 8, 6, 6, 49, 37, 27, 19, 13, 9, 7, 7, 64, 50, 38, 28, 20, 14, 10, 8, 8, 81, 65, 51, 39, 29, 21, 15, 11, 9, 9, 100, 82, 66, 52, 40, 30, 22, 16, 12, 10, 10, 121, 101, 83, 67, 53, 41, 31, 23, 17, 13

Offset: 0

Henry Bottomley, Jun 05 2000

			Table begins:
..0...1...4...9..16..25..36..49..64..81.100.121.144...
..1...2...5..10..17..26..37..50..65..82.101.122.145...
..2...3...6..11..18..27..38..51..66..83.102.123.146...
..3...4...7..12..19..28..39..52..67..84.103.124.147...
..4...5...8..13..20..29..40..53..68..85.104.125.148...
..5...6...9..14..21..30..41..54..69..86.105.126.149...
..6...7..10..15..22..31..42..55..70..87.106.127.150...
..7...8..11..16..23..32..43..56..71..88.107.128.151...
..8...9..12..17..24..33..44..57..72..89.108.129.152...
..9..10..13..18..25..34..45..58..73..90.109.130.153...
.10..11..14..19..26..35..46..59..74..91.110.131.154...
... - _Philippe Deléham_, Mar 31 2013

First column is A001477, second column is A000027, first row is A000290, second row is A002522, third row (apart from first term) is A010000, main diagonal is A002378, other diagonals include A028387, A028552, A014209, A002061, A014206, A027688-A027694, each row of A055096 (as upper right triangle) is right hand part of some row of this table

Cf. A185910

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A185911 Weight array of A185910, by antidiagonals.

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A185912 Accumulation array of A185910; by antidiagonals.

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A185913 Accumulation array of A185912, by antidiagonals.

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A055630 Table T(k,m) = k^2 + m read by antidiagonals.

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