%I A054523
%S A054523 1,1,1,2,0,1,2,1,0,1,4,0,0,0,1,2,2,1,0,0,1,6,0,0,0,0,0,1,4,2,0,
%T A054523 1,0,0,0,1,6,0,2,0,0,0,0,0,1,4,4,0,0,1,0,0,0,0,1,10,0,0,0,0,0,0,
%U A054523 0,0,0,1,4,2,2,2,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,1,6,6
%N A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise 
               (n >= 1, 1<=k<=n).
%C A054523 Comments from Gary Adamson, Jan 08 2007: (Start) Let H be this lower 
               triangular matrix. Then:
%C A054523 H * A051731 = A126988,
%C A054523 H * [1, 2, 3,...] = 1, 3, 5, 8, 9, 15,...= A018804,
%C A054523 H * sigma(n) = A038040 = d(n) * n = 1, 4, 6, 12, 10,... where sigma(n) 
               = A000203,
%C A054523 H * d(n) (A000005) = sigma(n) = A000203,
%C A054523 Row sums of H = A018804 = sum of GCD (k,n),
%C A054523 H^2 * d(n) = d(n)*n, H^2 = A127192,
%C A054523 H * mu(n) (A008683) = phi(n) = A000010,
%C A054523 H^2 row sums = A018804. (End)
%C A054523 The Mobius inversion principle of Richard Dedekind and Joseph Liouville 
               (1857). Cf. "Concrete Mathematics", p. 136; is equivalent to the 
               statement that row sums of triangle A054523 = n, where the triangle 
               = A054525 * A126988. A054525 = the Mobius transform and A126988 records 
               the divisors of n by rows. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 03 2008]
%C A054523 Row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,...}. [From Roger 
               L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 18 2008]
%D A054523 "Concrete Mathematics", Ronald L. Graham, Donald E. Knuth, & Oren Patashnik; 
               Addison-Wesley, 2-nd ed., 1994, p. 136. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 03 2008]
%F A054523 Equals A054525 * A126988 as infinite lower triangular matrices. [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 03 2008]
%F A054523 Can be obtained from the necklace polynomials in Mathematica: p(x,n)=(n/
               x)*NecklacePolynomial[n, x, Cyclic]]; t(n,m)=coefficients(p(x,n)). 
               [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 18 2008]
%e A054523 Triangle begins {1}, {1, 1}, {2, 0, 1}, {2, 1, 0, 1}, {4, 0, 0, 0, 1}, 
               {2, 2, 1, 0, 0, 1}, {6, 0, 0, 0, 0, 0, 1}, {4, 2, 0, 1, 0, 0, 0, 
               1}, {6, 0, 2, 0, 0, 0, 0, 0, 1}, {4, 4, 0, 0, 1, 0, 0, 0, 0, 1}, 
               {10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 
               0, 1} [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 18 2008]
%t A054523 << DiscreteMath`Combinatorica`; Table[ExpandAll[(n/x)*NecklacePolynomial[n, 
               x, Cyclic]], {n, 1, 12}]; Table[CoefficientList[ExpandAll[(n/x)*NecklacePolynomial[n, 
               x, Cyclic]], x], {n, 1, 12}]; Flatten[%] [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Nov 18 2008]
%Y A054523 Cf. A054521, A054525.
%Y A054523 Sequence in context: A117170 A117466 A136266 this_sequence A161363 A106351 
               A096800
%Y A054523 Adjacent sequences: A054520 A054521 A054522 this_sequence A054524 A054525 
               A054526
%K A054523 nonn,tabl
%O A054523 1,4
%A A054523 N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2000