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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

See illustration in links for construction rule.

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  ...

...

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

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: A055955 A165071 A223033 * A222918 A280480 A223104

Adjacent sequences:  A237441 A237442 A237443 * A237445 A237446 A237447

KEYWORD

sign,tabl

AUTHOR

Kival Ngaokrajang, Feb 08 2014

STATUS

approved

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Last modified December 2 16:46 EST 2020. Contains 338877 sequences. (Running on oeis4.)