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%I #4 Mar 30 2012 17:39:18
%S 1,2,1,5,0,2,6,3,0,2,17,0,0,0,4,10,5,4,0,0,2,37,0,0,0,0,0,6,22,11,0,6,
%T 0,0,0,4,41,0,14,0,0,0,0,0,6,34,17,0,0,8,0,0,0,0,4,101,0,0,0,0,0,0,0,
%U 0,0,10,30,15,12,10,0,6,0,0,0,0,0,4,145,0,0,0,0,0,0,0,0,0,0,0,12,74,37,0
%N Triangle T(n,k) read by rows: matrix product A054522 * A054523.
%C If the two matrices A054522 and A054523 are commuted, the matrix product becomes A127478.
%F T(n,k) = sum_{j=k..n} A054522(n,j) * A054523(j,k).
%F sum_{k=1..n} T(n,k) = A057660(n) (row sums).
%F T(n,n) = A000010(n) (diagonal).
%F T(n,1) = A029939(n).
%e First few rows of the triangle are:
%e 1;
%e 2, 1;
%e 5, 0, 2;
%e 6, 3, 0, 2;
%e 17, 0, 0, 0, 4;
%e 10, 5, 4, 0, 0, 2;
%e 37, 0, 0, 0, 0, 0, 6;
%e 22, 11, 0, 6, 0, 0, 0, 4;
%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 A127477 := proc(n,k) add( A054522(n,j)*A054523(j,k), j=k..n) ; end: seq(seq( A127477(n,k),k=1..n),n=1..15) ;
%Y Cf. A054522, A054523, A057660, A000010, A029939.
%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
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