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A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

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