%I #4 Mar 30 2012 17:39:18
%S 1,2,1,3,0,2,4,2,0,2,5,0,0,0,4,6,3,4,0,0,2,7,0,0,0,0,0,6,8,4,0,4,0,0,
%T 0,4,9,0,6,0,0,0,0,0,6,10,5,0,0,8,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,10,
%U 12,6,8,6,0,4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,0,0,0,0,12,14,7,0,0,0,0,12,0,0,0
%N Triangle T(n,k) read by rows: matrix product A054523 * A054522.
%C If the two matrices A054523 and A054522 are commuted, the matrix product becomes A127477.
%F T(n,k) = sum_{j=k..n} A054523(n,j) * A054522(j,k).
%F T(n,n) = A000010(n) (diagonal).
%F sum_{k=1..n} T(n,k) = A018804(n) (row sums).
%e First few rows of the triangle are:
%e .1;
%e .2, 1;
%e .3, 0, 2;
%e .4, 2, 0, 2;
%e .5, 0, 0, 0, 4;
%e .6, 3, 4, 0, 0, 2;
%e .7, 0, 0, 0, 0, 0, 6;
%e .8, 4, 0, 4, 0, 0, 0, 4;
%e ....
%p A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
%p A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
%p A127478 := proc(n,k) add( A054523(n,j)*A054522(j,k), j=k..n) ; end: seq(seq( A127478(n,k),k=1..n),n=1..15) ;
%Y Cf. A054522, A054523, A018804, A000010.
%K nonn,tabl,easy
%O 1,2
%A _Gary W. Adamson_, Jan 15 2007
%E Converted comments to formulas, extended - _R. J. Mathar_, Sep 11 2009