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Given (-1)*triangle A077049, preface this with a "1" as row 1; = M.

Perform M * A128407 (the diagonalized variant of A008683); = A176918 as an

infinite lower triangular matrix.

As infinite lower triangular matrices, A077049 * the diagonalized version of

A002033: (1, 1, 1, 2, 1, 3, 1, 4, 2,...) as the right border with the rest zeros.

{T(n, k)*k, k=1..n} setminus {0} = divisors(n).

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.

Sum_{k=1..n} T(n, k) = A000005(n).

Sum_{k=1..n} T(n, k)*k = A000203(n).

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 = 1101_2; row 4 = 1, 1, 0, 1. - Philippe Deléham, Oct 04 2003

From Paul Barry, Dec 05 2004: (Start)

Binomial transform (product by binomial matrix) is A101508.

Columns have g.f.: x^k/(1-x^(k+1)) (k >= 0). (End)

Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/k) if k|n, 0 otherwise. - Paul D. Hanna, Jan 09 2006

From Gary W. Adamson, Apr 15 2007, May 10 2007: (Start)

Equals A129372 * A115361 as infinite lower triangular matrices.

A054525 is the inverse of this triangle (as lower triangular matrix).

This triangle * [1, 2, 3, ...] = sigma(n) (A000203).

This triangle * [1/1, 1/2, 1/3, ...] = sigma(n)/n. (End)

From Reinhard Zumkeller, Nov 01 2009: (Start)

T(n, k) = 0^(n mod k).

T(n, k) = A000007(A048158(n, k)). (End)

From Mats Granvik, Jan 26 2010, Feb 10 2010, Feb 16 2010: (Start)

T(n, k) = A172119(n) mod 2.

T(n, k) = A175105(n) mod 2.

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.

(Jeffrey O. Shallit kindly provided a clarification along with a proof of this formula.) (End)

A049820(n) = number of zeros in n-th row. - Reinhard Zumkeller, Mar 09 2010

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

T(n, k) = Sum_{y=1..n} Sum_{x=1..n} [GCD((x/y)*(k/n), n) = k]. - Mats Granvik, Dec 17 2023

a(i,j) = 1 if j=1 or i|j; 0 otherwise.

a(A000217(n)) = a(A000217(n)+1) = 1. - Enrique Pérez Herrero, Apr 16 2010

T(n,k) = c_n(k) = phi(n) * Moebius(n/gcd(n, k))/phi(n/gcd(n, k)). - Emeric Deutsch, Dec 23 2004 [The r.h.s. of this formula is known as the von Sterneck function, and it was introduced by him around 1900. - Petros Hadjicostas, Jul 20 2019]

Dirichlet series: Sum_{n>=1} c_n(k)/n^s = sigma_{1-s}(k)/zeta(s) where sigma is the sum-of-divisors function. Sum_{n>=1} c_k(n)/n^s = zeta(s)*Sum_{d|k} mu(k/d)*d^(1-s). [Hardy & Wright, Titchmarsh] - R. J. Mathar, Apr 01 2012 [We have sigma_{1-s}(k) = Sum_{d|k} d^{1-s} = Sum_{d|k} (k/d)^{1-s} = sigma_{s-1}(k) / k^{s-1}. - Petros Hadjicostas, Jul 27 2019]

From Mats Granvik, Oct 10 2016: (Start)

For n >= 1 and k >= 1 let

A(n,k) := if n mod k = 0 then k^r, otherwise 0;

B(n,k) := if n mod k = 0 then k/n^s, otherwise 0.

Then the Ramanujan's sum matrix equals

inverse(A).transpose(B) evaluated at s=0 and r=0.

Equals inverse(A051731).transpose(A127093).

Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} T(n,k)/(n^r*k^s) = zeta(s)*zeta(s + r - 1)/zeta(r) as in Wikipedia. (End)

T(n,k) = c_n(k) = Sum_{s | gcd(n,k)} s * Moebius(n/s). - Petros Hadjicostas, Jul 27 2019

Lambert series and a consequence: Sum_{n >= 1} c_n(k) * z^n / (1 - z^n) = Sum_{s|k} s * z^s and -Sum_{n >= 1} (c_n(k) / n) * log(1 - z^n) = Sum_{s|k} z^s for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 15 2019

From Franklin T. Adams-Watters, Jan 28 2006: (Start)

T(n,k) = Sum_{i=1..k} A077049(n,i).

G.f.: (1/(1-x))*Sum_{k>0} x^k*y^k/(1-x^k) = (1/(1-x))*Sum_{k>0} x^k * y / (1 - x^k y) = (1/(1-x)) * Sum_{k>0} x^k * Sum_{d|k} y^d = A(x,y)/(1-x) where A(x,y) is the g.f. of A077049. (End)

T(n,k) = floor((n + 1 - k) / k). - Reinhard Zumkeller, Apr 13 2012

T(n, k)=1 if n|k else T(n, k)=0.

From Boris Putievskiy, May 08 2013: (Start)

As table T(n,k)= floor(n/k)-floor((n-1)/k).

As linear sequence a(n) = floor(A002260(n)/A004736(n)) - floor((A002260(n)-1)/A004736(n));

a(n) = floor(i/j) - floor((i-1)/j), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

G.f. for row n: x^n/(1-x^n). - Geoffrey Critzer, Mar 29 2015

M = T^(-1), where T is the left summatory matrix, A077049.

R=U*V, where U and V are the summatory matrices (A077049, A077051). The triangle T(n, k) formed by antidiagonals: T(n, k)=tau(gcd(k, n+1-k)) for 1<=k<=n, where tau(m)=A000005(m). [Corrected by Leroy Quet, Apr 08 2009]

Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} tau(gcd(n,k))/n^s/k^c = zeta(s)*zeta(c)* zeta(s + c). - Mats Granvik, May 19 2021

T(n,k) = 1 if n=1; otherwise, if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.