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A127096 Triangle T(n,m) = A000012*A127094 read by rows.

%I A127096 #20 Sep 15 2023 05:26:00
%S A127096 1,3,1,6,1,1,10,1,3,1,15,1,3,1,1,21,1,3,4,3,1,28,1,3,4,3,1,1,36,1,3,4,
%T A127096 7,1,3,1,45,1,3,4,7,1,6,1,1,55,1,3,4,7,6,6,1,3,1,66,1,3,4,7,6,6,1,3,1,
%U A127096 1,78,1,3,4,7,6,12,1,7,4,3,1,91,1,3,4,7,6,12,1,7,4,3,1,1,105,1,3,4,7,6,12,8,7,4,3,1,3,1
%N A127096 Triangle T(n,m) = A000012*A127094 read by rows.
%C A127096 Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127094 from the right.
%F A127096 T(n,m) = Sum_{j=m..n} A000012(n,j)*A127094(j,m) = Sum_{j=m..n} A127094(j,m).
%e A127096 First few rows of the triangle are:
%e A127096    1;
%e A127096    3, 1,
%e A127096    6, 1, 1;
%e A127096   10, 1, 3, 1;
%e A127096   15, 1, 3, 1, 1;
%e A127096   21, 1, 3, 4, 3, 1;
%e A127096   28, 1, 3, 4, 3, 1, 1;
%e A127096   ...
%p A127096 A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
%p A127096 A127094 := proc(n,m) A127093(n, n-m+1) ; end:
%p A127096 A127096 := proc(n,m) add( A127094(j,m),j=m..n) ; end:
%p A127096 for n from 1 to 15 do for m from 1 to n do printf("%d,",A127096(n,m)) ; od: od: # _R. J. Mathar_, Aug 18 2009
%t A127096 T[n_, m_] := Sum[1 + Mod[j, m - j - 1] - Mod[1 + j, m - j - 1], {j, m, n}];
%t A127096 Table[T[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 15 2023 *)
%Y A127096 Cf. A127093, A127094, A123229, A024916 (row sums), A000203, A126988.
%K A127096 nonn,easy,tabl
%O A127096 1,2
%A A127096 _Gary W. Adamson_, Jan 05 2007
%E A127096 Edited and extended by _R. J. Mathar_, Aug 18 2009