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A051731
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Triangle read by rows: T(n,k)=1 if k divides n, T(n,k)=0 otherwise.
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232
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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, 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, 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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| {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
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
A054525 = the inverse of this triangle = A129360 * A115369. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
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]
T(n,k) = (1-((-1)^A175105))/2. [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 10 2010]
A049820(n) = number of zeros in n-th row. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 09 2010]
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LINKS
| Mats Granvik, Illustration of A051731
Jeffrey Ventrella, Divisor Plot [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 08 2009]
Mats Granvik, Better illustration of A051731
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FORMULA
| 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
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
Columns have g.f. x^k/(1-x^(k+1)) (k>=0). - Paul Barry (pbarry(AT)wit.ie), Dec 05 2004
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
Equals = A129372 * A115361 as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
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
T(n,k) = 0^(n mod k). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 01 2009]
T(n,k) = (1-((-1)^A172119))/2. [From Mats Granvik (mats.granvik(AT)abo.fi), Jan 26 2010]
Jeffrey O. Shallit kindly provided a clarification along with a proof of this formula T(n,1)=1, k>1: T(n,k) = (sum from i = 1 to k-1 of T(n-i,k-1)) - (sum from i = 1 to k-1 of T(n-i,k)). [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 16 2010]
T(n,k)=(A181116/A181117)*(A181116/A181117). [From Mats Granvik (mats.granvik(AT)abo.fi), Oct 04 2010]
T(n,k)= limit as r -> infinity of ((cos(2*pi*n/k)+1)/2)^r.
T(n,k)= limit as r -> infinity of (1-(sin(pi*n/k)/(pi*n/k))^2)^r. Setting r = 1 and k = 1 the expression is the same as the term in the Montgomery pair correlation conjecture. [From Mats Granvik, Dec 09 2011]
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EXAMPLE
| Triangle begins:
.{1};
.{1,1};
.{1,0,1};
.{1,1,0,1};
.{1,0,0,0,1}; ...
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MATHEMATICA
| Flatten[Table[If[Mod[n, k] == 0, 1, 0], {n, 20}, {k, n}]]
Clear[t]; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}], 0]; Flatten[Table[t[n, k], {n, 12}, {k, n}]] (* Mats Granvik, Jan 23 2012 *)
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CROSSREFS
| Cf. A000005, A000203, A074854, A054525, A129372, A115361.
A077049 and A077051 are other presentations of this matrix. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 08 2009]
T(n,k) = A000007(A048158(n,k)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 01 2009]
Sequence in context: A110471 A174854 A103994 * A135839 A071022 A155076
Adjacent sequences: A051728 A051729 A051730 * A051732 A051733 A051734
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KEYWORD
| easy,nice,nonn,tabl
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AUTHOR
| Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
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