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A225233 Triangle read by rows: T(n, k) = (2*n + 2 - k)*k, for 0 <= k <= n.

%I A225233 #22 Jun 02 2021 15:08:02
%S A225233 0,0,3,0,5,8,0,7,12,15,0,9,16,21,24,0,11,20,27,32,35,0,13,24,33,40,45,
%T A225233 48,0,15,28,39,48,55,60,63,0,17,32,45,56,65,72,77,80,0,19,36,51,64,75,
%U A225233 84,91,96,99,0,21,40,57,72,85,96,105,112,117,120,0,23,44,63,80,95,108,119,128,135,140,143,0,25
%N A225233 Triangle read by rows: T(n, k) = (2*n + 2 - k)*k, for 0 <= k <= n.
%C A225233 The entries of a row n appear on the diagonal of a square array of dimension n + 1 while filling it with numbers from 0 to n^2 - 1 first along top row and left column, then along 2nd row and 2nd column, 3rd row and 3rd column etc. up to the (single) entry in the n-th row and n-th column. [This may be the preferred order if a set of matrices M is built with requirements on the product M*M.] This vaguely is an alternative to the boustrophedonic re-arrangement of a finite array.
%C A225233 The triangle may also be generated by reading half of each second antidiagonal of the array A003991.
%C A225233 The numbers appear in reverse order as the numerators in the triangle A061035 before they are reduced with the denominators by cancellation of common factors. - _Paul Curtz_, May 03 2013
%e A225233 Triangle starts:
%e A225233 [0] 0;
%e A225233 [1] 0,  3;
%e A225233 [2] 0,  5,  8;
%e A225233 [3] 0,  7, 12, 15;
%e A225233 [4] 0,  9, 16, 21, 24;
%e A225233 [5] 0, 11, 20, 27, 32, 35;
%e A225233 [6] 0, 13, 24, 33, 40, 45, 48;
%e A225233 [7] 0, 15, 28, 39, 48, 55, 60, 63;
%e A225233 [8] 0, 17, 32, 45, 56, 65, 72, 77, 80;
%e A225233 [9] 0, 19, 36, 51, 64, 75, 84, 91, 96, 99.
%e A225233 .
%e A225233 The row n = 3, for example, is created by reading the 4 X 4 square array downwards its main diagonal.
%e A225233 0,  1,  3,  5;
%e A225233 2,  7,  8, 10;
%e A225233 4,  9, 12, 13;
%e A225233 6, 11, 14, 15;
%p A225233 T := proc(n, k) option remember; if k = 0 then 0 elif k = 1 then 2*n+1 else
%p A225233 T(n, k-1) + T(n-k+1, 1) fi end:
%p A225233 for n from 0 to 9 do seq(T(n, k), k=0..n) od; # _Peter Luschny_, Jun 02 2021
%Y A225233 Cf. A016061 (row sums), A045944 (central), A005563 (main diagonal).
%Y A225233 Cf. A005408 (column k=2), A008586 (column k=3), A016945 (column k=4).
%Y A225233 Cf. A003991, A061035.
%K A225233 easy,nonn,tabl
%O A225233 0,3
%A A225233 _R. J. Mathar_, May 03 2013
%E A225233 Offset set to 0 and edited by _Peter Luschny_, Jun 02 2021