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%I #31 Sep 03 2015 15:06:32
%S 0,1,0,1,1,0,4,2,1,0,4,5,3,1,0,9,7,6,4,1,0,9,12,10,7,5,1,0,16,15,15,
%T 13,8,6,1,0,16,22,21,18,16,9,7,1,0,25,26,28,27,21,19,10,8,1,0,25,35,
%U 36,34,33,24,22,11,9,1,0,36,40,45,46,40,39,27,25,12,10,1,0
%N Square array read by antidiagonals with T(n,k) = n*((k+2)*n-k)/2, n=0, +- 1, +- 2,..., k>=0.
%C Also, column k lists the partial sums of the column k of A195151. The first differences in row n are always the n-th term of the triangular numbers repeated 0,0,1,1,3,3,6,6,... ([0,0] together with A008805).
%C Also, for k >= 1, this is a table of generalized polygonal numbers since column k lists the generalized m-gonal numbers, where m = k+4, for example: if k = 1 then m = 5, so the column 1 lists the generalized pentagonal numbers A001318 (see example).
%F T(n,k) = (k+2)*n*(n+1)/8+(k-2)*((2*n+1)*(-1)^n-1)/16, n >= 0 and k >= 0. - _Omar E. Pol_, Oct 01 2011
%e Array begins:
%e . 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%e . 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
%e . 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...
%e . 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,...
%e . 4, 7, 10, 13, 16, 19, 22, 25, 28, 31,...
%e . 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,...
%e . 9, 15, 21, 27, 33, 39, 45, 51, 57, 63,...
%e . 16, 22, 28, 34, 40, 46, 52, 58, 64, 70,...
%e . 16, 26, 36, 46, 56, 66, 76, 86, 96, 106,...
%e . 25, 35, 45, 55, 65, 75, 85, 95, 105, 115,...
%e . 25, 40, 55, 70, 85, 100, 115, 130, 145, 160,...
%e ...
%Y Rows: A000004, A000012, A000027.
%Y Column 0 gives A008794, except its first term.
%Y Columns >= 1: A001318, (A000217), A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818.
%Y Cf. A008805, A152151.
%K nonn,tabl
%O 0,7
%A _Omar E. Pol_, Sep 14 2011
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