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A173997 Irregular triangle by columns derived from (1, 2, 3, ...) * (1, 2, 3, ...).

%I A173997 #27 May 13 2022 09:59:38
%S A173997 1,2,3,2,4,4,5,6,3,6,8,6,7,10,9,4,8,12,12,8,9,14,15,12,5,10,16,18,16,
%T A173997 10,11,18,21,20,15,6,12,20,24,24,20,12,13,22,27,28,25,18,7,14,24,30,
%U A173997 32,30,24,14,15,26,33,36,35,30,21,8,16,28,36,40,40,36,28,16
%N A173997 Irregular triangle by columns derived from (1, 2, 3, ...) * (1, 2, 3, ...).
%C A173997 Given a (1, 2, 3, ...) * (1, 2, 3, ...) multiplication table; leftmost column of the triangle = (1, 2, 3, ...). Then shift down each successive column of the array twice to get this irregular triangle.
%H A173997 Stefano Spezia, <a href="/A173997/b173997.txt">First 200 rows of the triangle, flattened</a>
%F A173997 T(n, k) = k*(2 - 2*k + n), with 1 <= k <= floor((n + 1)/2). - _Stefano Spezia_, Apr 19 2022
%e A173997 Given:
%e A173997   1,  2,  3,  4,  5, ...
%e A173997   2,  4,  6,  8, 10, ...
%e A173997   3,  6,  9, 12, 15, ...
%e A173997   4,  8, 12, 16, 20, ...
%e A173997   ...
%e A173997 After the shift twice operation, we obtain:
%e A173997    1;
%e A173997    2;
%e A173997    3,  2;
%e A173997    4,  4;
%e A173997    5,  6,  3;
%e A173997    6,  8,  6;
%e A173997    7, 10,  9,  4;
%e A173997    8, 12, 12,  8;
%e A173997    9, 14, 15, 12,  5;
%e A173997   10, 16, 18, 16, 10;
%e A173997   11, 18, 21, 20, 15,  6;
%e A173997   12, 20, 24, 24, 20, 12;
%e A173997   ...
%t A173997 Flatten[Table[k(2-2k+n),{n,16},{k,Floor[(n+1)/2]}]] (* _Stefano Spezia_, Apr 19 2022 *)
%Y A173997 Cf. A003991, A006918 (row sums).
%K A173997 nonn,easy,tabf
%O A173997 1,2
%A A173997 _Gary W. Adamson_, Mar 05 2010