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%I #137 Dec 26 2023 11:24:05
%S 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,
%T 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,
%U 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
%N Triangle read by rows: T(n, k) = 1 if k divides n, T(n, k) = 0 otherwise, for 1 <= k <= n.
%C T(n, k) is the number of partitions of n into k equal parts. - _Omar E. Pol_, Apr 21 2018
%C This triangle is the lower triangular array L in the LU decomposition of the square array A003989. - _Peter Bala_, Oct 15 2023
%H Charles R Greathouse IV, <a href="/A051731/b051731.txt">Rows n = 1..100, flattened</a>
%H Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.
%H Mats Granvik, <a href="/A051731/a051731_1.gif">Illustration</a>.
%H Warren P. Johnson, <a href="https://doi.org/10.1080/0025570X.2003.11953215">An LDU Factorization in Elementary Number Theory</a>, Mathematics Magazine, Vol. 76, No. 5 (Dec., 2003), pp. 392-394.
%H Jeffrey Ventrella, <a href="http://www.divisorplot.com/">Divisor Plot</a>.
%F {T(n, k)*k, k=1..n} setminus {0} = divisors(n).
%F Sum_{k=1..n} T(n, k)*k^i = sigma[i](n), where sigma[i](n) is the sum of the i-th power of the positive divisors of n.
%F Sum_{k=1..n} T(n, k) = A000005(n).
%F Sum_{k=1..n} T(n, k)*k = A000203(n).
%F 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 Rows given by A074854 converted to binary. Example: A074854(4) = 13 = 1101_2; row 4 = 1, 1, 0, 1. - _Philippe Deléham_, Oct 04 2003
%F From _Paul Barry_, Dec 05 2004: (Start)
%F Binomial transform (product by binomial matrix) is A101508.
%F Columns have g.f.: x^k/(1-x^(k+1)) (k >= 0). (End)
%F Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/k) if k|n, 0 otherwise. - _Paul D. Hanna_, Jan 09 2006
%F From _Gary W. Adamson_, Apr 15 2007, May 10 2007: (Start)
%F Equals A129372 * A115361 as infinite lower triangular matrices.
%F A054525 is the inverse of this triangle (as lower triangular matrix).
%F This triangle * [1, 2, 3, ...] = sigma(n) (A000203).
%F This triangle * [1/1, 1/2, 1/3, ...] = sigma(n)/n. (End)
%F From _Reinhard Zumkeller_, Nov 01 2009: (Start)
%F T(n, k) = 0^(n mod k).
%F T(n, k) = A000007(A048158(n, k)). (End)
%F From _Mats Granvik_, Jan 26 2010, Feb 10 2010, Feb 16 2010: (Start)
%F T(n, k) = A172119(n) mod 2.
%F T(n, k) = A175105(n) mod 2.
%F T(n, k) = Sum_{i=1..k-1} (T(n-i, k-1) - T(n-i, k)) for k > 1 and T(n, 1) = 1.
%F (Jeffrey O. Shallit kindly provided a clarification along with a proof of this formula.) (End)
%F A049820(n) = number of zeros in n-th row. - _Reinhard Zumkeller_, Mar 09 2010
%F The determinant of this matrix where T(n, n) has been swapped with T(1,k) is equal to the n-th term of the Mobius function. - _Mats Granvik_, Jul 21 2012
%F T(n, k) = Sum_{y=1..n} Sum_{x=1..n} [GCD((x/y)*(k/n), n) = k]. - _Mats Granvik_, Dec 17 2023
%e The triangle T(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e 1: 1
%e 2: 1 1
%e 3: 1 0 1
%e 4: 1 1 0 1
%e 5: 1 0 0 0 1
%e 6: 1 1 1 0 0 1
%e 7: 1 0 0 0 0 0 1
%e 8: 1 1 0 1 0 0 0 1
%e 9: 1 0 1 0 0 0 0 0 1
%e 10: 1 1 0 0 1 0 0 0 0 1
%e 11: 1 0 0 0 0 0 0 0 0 0 1
%e 12: 1 1 1 1 0 1 0 0 0 0 0 1
%e 13: 1 0 0 0 0 0 0 0 0 0 0 0 1
%e 14: 1 1 0 0 0 0 1 0 0 0 0 0 0 1
%e 15: 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1
%e ... Reformatted and extended. - _Wolfdieter Lang_, Nov 12 2014
%p A051731 := proc(n, k) if n mod k = 0 then 1 else 0 end if end proc:
%p # _R. J. Mathar_, Jul 14 2012
%t Flatten[Table[If[Mod[n, k] == 0, 1, 0], {n, 20}, {k, n}]]
%o (PARI)
%o for(n=1,9,for(k=1,n,print1(!(n%k)", "))) \\ _Charles R Greathouse IV_, Mar 14 2012
%o (Haskell)
%o a051731 n k = 0 ^ mod n k
%o a051731_row n = a051731_tabl !! (n-1)
%o a051731_tabl = map (map a000007) a048158_tabl
%o -- _Reinhard Zumkeller_, Aug 13 2013
%o (Sage)
%o A051731_row = lambda n: [int(k.divides(n)) for k in (1..n)]
%o for n in (1..15): print(A051731_row(n)) # _Peter Luschny_, Jan 05 2018
%Y Variants: A113704, A077049, A077051.
%Y Cf. A000005 (row sums), A032741(n+2) (diagonal sums), A243987 (partial sums per row).
%Y Cf. A000203, A003989, A074854, A054525, A129372, A115361, A002260.
%Y Cf. A134546 (A004736 * T, matrix multiplication).
%K easy,nice,nonn,tabl
%O 1,1
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%E Edited by _Peter Luschny_, Oct 18 2023
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