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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.

%I A237444 #17 Nov 02 2021 06:32:57
%S A237444 0,1,-1,6,0,-6,18,6,-6,-18,40,20,0,-20,-40,75,45,15,-15,-45,-75,126,
%T A237444 84,42,0,-42,-84,-126,196,140,84,28,-28,-84,-140,-196,288,216,144,72,
%U A237444 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
%N 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.
%C A237444 See illustration in links for construction rule.
%C A237444 Column 1 = A002411.
%C A237444 Column 2 = A005564 ,for n >= 3.
%C A237444 Column 3 first differences = A140091.
%C A237444 Nonnegative numbers of this sequence are given by A082375(n,k)*A000217(n), (see example). - _Philippe Deléham_, Feb 08 2014
%H A237444 Kival Ngaokrajang, <a href="/A237444/a237444.pdf">Illustration for n = 1..5</a>
%F A237444 T(n,k) = - T(n,n-k+1), T(2n+1,n+1)= 0. - _Philippe Deléham_, Feb 08 2014
%F A237444 T(n+1,k+1) = A114327(n,k)*A000217(n). - _Philippe Deléham_, Feb 08 2014
%e A237444 Triangle begins:
%e A237444 n/k   1   2   3   4  5    6   7    8    9   ...
%e A237444 1   0
%e A237444 2   1  -1
%e A237444 3   6   0  -6
%e A237444 4  18   6  -6  18
%e A237444 5  40  20   0 -20 -40
%e A237444 6  75  45  15 -15 -45 -75
%e A237444 7 126  84  42   0 -42 -84 -126
%e A237444 8 196 140  84  28 -28 -84 -140 -196
%e A237444 9 288 216 144  72   0 -72 -144 -216 -288  ...
%e A237444 ...
%e A237444 A082375 begins:
%e A237444 0;
%e A237444 1;
%e A237444 2, 0;
%e A237444 3, 1;
%e A237444 4, 2, 0;
%e A237444 5, 3, 1;
%e A237444 6, 4, 2, 0;
%e A237444 7, 5, 3, 1;
%e A237444 8, 6, 4, 2, 0;
%e A237444 9, 7, 5, 3, 1;
%e A237444 .....
%e A237444 A000217 (triangular numbers) begins:
%e A237444 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
%e A237444 A082375(n,k)*A000217(n) begins:
%e A237444 0;
%e A237444 1;
%e A237444 6, 0;
%e A237444 18, 6;
%e A237444 40, 20, 0;
%e A237444 75, 45, 15;
%e A237444 126, 84, 42, 0;
%e A237444 196, 140, 84, 28;
%e A237444 288, 216, 144, 72, 0;
%e A237444 405, 315, 225, 135, 45;
%e A237444 ... - _Philippe Deléham_, Feb 08 2014
%o A237444 (Small Basic)
%o A237444 For n = 1 to 20
%o A237444 For n1 = 1 To n
%o A237444 c = 0
%o A237444 r = 0
%o A237444   For n2 = 1+n*(n1-1) To n+n*(n1-1)
%o A237444     c = c + n2
%o A237444   Endfor
%o A237444   For n3 = n1 To n1+n*(n-1) Step n
%o A237444     r = r + n3
%o A237444   EndFor
%o A237444 a = r - c
%o A237444 TextWindow.Write(a+", ")
%o A237444 EndFor
%o A237444 EndFor
%Y A237444 Cf. A000217, A002411, A005564, A114327, A140091.
%K A237444 sign,tabl
%O A237444 1,4
%A A237444 _Kival Ngaokrajang_, Feb 08 2014