Search: id:A051731
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%I A051731
%S A051731 1,1,1,1,0,1,1,1,0,1,1,0,0,0,1,1,1,1,0,0,1,1,0,0,0,0,0,1,1,1,0,1,0,0,0,
%T A051731 1,1,0,1,0,0,0,0,0,1,1,1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,
%U A051731 0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1
%N A051731 Triangle read by rows: T(n,k)=1 if k divides n, T(n,k)=0 otherwise.
%C A051731 {T(n,k)*k, k=1..n} setminus {0} = divisors of n; sum(T(n,k)*(k^i),k=1..n) 
               = sigma[i](n) = sum of the i-th power of positive divisors of n; 
               sum(T(n,k),k=1..n)=A000005, sum(T(n,k)*k,k=1..n)=A000203
%C A051731 Row sums are A000005. Diagonal sums are A032741(n+2). Might be called 
               a Mobius matrix. Binomial transform (product by binomial matrix) 
               is A101508. - Paul Barry (pbarry(AT)wit.ie), Dec 05 2004
%C A051731 A054525 = the inverse of this triangle = A129360 * A115369. - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
%C A051731 If the 1 in the lower right corner is moved to the upper right corner 
               then the determinant gives the mobius function. [From Mats Granvik 
               (mats.granvik(AT)abo.fi), Nov 18 2008]
%H A051731 Mats Granvik, <a href="a051731.gif">Illustration of A051731</a>
%H A051731 Jeffrey Ventrella, <a href="http://www.divisorplot.com/">Divisor Plot</
               a> [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 08 2009]
%F A051731 T(n, k)=T(n-k, k) for k<=n/2, T(n, k)=0 for n/2<k<=n-1, T(n, n)=1
%F A051731 Rows given by A074854 converted to binary. Example: A074854(4)= 13(decimal)= 
               1101(binary); row 4 = 1, 1, 0, 1. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Oct 04 2003
%F A051731 Columns have g.f. x^k/(1-x^(k+1)) (k>=0). - Paul Barry (pbarry(AT)wit.ie), 
               Dec 05 2004
%F A051731 Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/
               k) if k|n, 0 otherwise. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Jan 09 2006
%F A051731 Equals = A129372 * A115361 as infinite lower triangular matrices. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
%F A051731 This triangle * [1,2,3,...] = Sigma(n), A000203: (1, 3, 4, 7, 6, 12, 
               8,...). A051731 * [1/1, 1/2, 1/3,...] = Sigma(n)/n: (1/1, 3/2, 4/
               3, 7/4, 6/5,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 10 
               2007
%F A051731 T(n,k) = 0^(n mod k). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 01 2009]
%e A051731 Triangle begins:
%e A051731 .{1};
%e A051731 .{1,1};
%e A051731 .{1,0,1};
%e A051731 .{1,1,0,1};
%e A051731 .{1,0,0,0,1}; ...
%Y A051731 Cf. A000005, A000203, A074854, A054525, A129372, A115361.
%Y A051731 A077049 and A077051 are other presentations of this matrix. [From Franklin 
               T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 08 2009]
%Y A051731 T(n,k) = A000007(A048158(n,k)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 01 2009]
%Y A051731 Sequence in context: A054524 A110471 A103994 this_sequence A135839 A155076 
               A120529
%Y A051731 Adjacent sequences: A051728 A051729 A051730 this_sequence A051732 A051733 
               A051734
%K A051731 easy,nice,nonn,tabl,new
%O A051731 1,1
%A A051731 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

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