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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.
1
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
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
KEYWORD
sign,tabl
AUTHOR
Kival Ngaokrajang, Feb 08 2014
STATUS
approved