A195152 Square array read by antidiagonals with T(n,k) = n*((k+2)*n-k)/2, n=0, +- 1, +- 2,..., k>=0.

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, 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, 36, 34, 33, 24, 22, 11, 9, 1, 0, 36, 40, 45, 46, 40, 39, 27, 25, 12, 10, 1, 0

Offset: 0

Author

Omar E. Pol, Sep 14 2011

Keywords

• nonn
• tabl

Comments

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).

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).

Examples

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,...
.  4,   5,   6,   7,   8,   9,  10,  11,  12,  13,...
.  4,   7,  10,  13,  16,  19,  22,  25,  28,  31,...
.  9,  12,  15,  18,  21,  24,  27,  30,  33,  36,...
.  9,  15,  21,  27,  33,  39,  45,  51,  57,  63,...
. 16,  22,  28,  34,  40,  46,  52,  58,  64,  70,...
. 16,  26,  36,  46,  56,  66,  76,  86,  96, 106,...
. 25,  35,  45,  55,  65,  75,  85,  95, 105, 115,...
. 25,  40,  55,  70,  85, 100, 115, 130, 145, 160,...
...
		

Crossrefs

Rows: A000004, A000012, A000027.

Column 0 gives A008794, except its first term.

Columns >= 1: A001318, (A000217), A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818.

Cf. A008805, A152151.

Formula

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