A237444 Triangle read by rows, T(n,k) is difference of column sum and row sum of natural numbers filled in n x n square.

0, 1, -1, 6, 0, -6, 18, 6, -6, -18, 40, 20, 0, -20, -40, 75, 45, 15, -15, -45, -75, 126, 84, 42, 0, -42, -84, -126, 196, 140, 84, 28, -28, -84, -140, -196, 288, 216, 144, 72, 0, -72, -144, -216, -288, 405, 315, 225, 135, 45, -45, -135, -225, -315, -405, 550, 440, 330, 220, 110, 0, -110, -220, -330, -440, -550, 726, 594, 462, 330, 198, 66, -66

Offset

1,4

Comments

See illustration in links for construction rule.

Column 1 = A002411.

Column 2 = A005564 ,for n >= 3.

Column 3 first differences = A140091.

Nonnegative numbers of this sequence are given by A082375(n,k)*A000217(n), (see example). - Philippe Deléham, Feb 08 2014

Links

Table of n, a(n) for n=1..73.

Kival Ngaokrajang, Illustration for n = 1..5

Formula

T(n,k) = - T(n,n-k+1), T(2n+1,n+1)= 0. - Philippe Deléham, Feb 08 2014

T(n+1,k+1) = A114327(n,k)*A000217(n). - Philippe Deléham, Feb 08 2014

Example

Triangle begins:
n/k   1   2   3   4  5    6   7    8    9   ...
1   0
2   1  -1
3   6   0  -6
4  18   6  -6  18
5  40  20   0 -20 -40
6  75  45  15 -15 -45 -75
7 126  84  42   0 -42 -84 -126
8 196 140  84  28 -28 -84 -140 -196
9 288 216 144  72   0 -72 -144 -216 -288  ...
...
A082375 begins:
0;
1;
2, 0;
3, 1;
4, 2, 0;
5, 3, 1;
6, 4, 2, 0;
7, 5, 3, 1;
8, 6, 4, 2, 0;
9, 7, 5, 3, 1;
.....
A000217 (triangular numbers) begins:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
A082375(n,k)*A000217(n) begins:
0;
1;
6, 0;
18, 6;
40, 20, 0;
75, 45, 15;
126, 84, 42, 0;
196, 140, 84, 28;
288, 216, 144, 72, 0;
405, 315, 225, 135, 45;
... - Philippe Deléham, Feb 08 2014

Prog

(Small Basic)
For n = 1 to 20
For n1 = 1 To n
c = 0
r = 0
  For n2 = 1+n*(n1-1) To n+n*(n1-1)
    c = c + n2
  Endfor
  For n3 = n1 To n1+n*(n-1) Step n
    r = r + n3
  EndFor
a = r - c
TextWindow.Write(a+", ")
EndFor
EndFor

Crossrefs

Cf. A000217, A002411, A005564, A114327, A140091.

Sequence in context: A165071 A341483 A223033 * A222918 A280480 A223104

Adjacent sequences: A237441 A237442 A237443 * A237445 A237446 A237447

Keyword

sign,tabl

Author

Kival Ngaokrajang, Feb 08 2014

Status

approved