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A191898 Symmetric square array read by anti-diagonals. T(n,1)=1, T(1,k)=1, n>=k: -Sum from i=1 to k-1 of T(n-i,k), n<k: -Sum from i=1 to n-1 of T(k-i,n). 16
1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -2, 1, 1, -2, 1, 1, 1, -1, 1, -1, -4, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -6, -1, 1, -1, 1, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

Rows equal columns and are periodic. No zero elements are found (conjecture). The recurrence is related to the recurrence for the Mahonian numbers. The main diagonal is the Dirichlet inverse of the Euler totient function (conjecture).

The sums from n=1 to infinity of T(n,k)/n converge to the Mangoldt function for column k (conjecture).

If gcd(n,k)=1 then T(n,k)=1 and if T(n,k)=1 then gcd(n,k)=1 (conjecture).

The Dirichlet generating functions for s>1 for the columns appear to be (see A054535):

Zeta(s)*(1 + (Sum over all possible combinations of products of negative distinct prime factors of k, up to rearrangement, 1/((-1* first distinct prime factor)*(-1*second distinct prime factor)*(-1*third distinct prime factor * ...))^(s-1))

Examples:

k=1: Zeta(s)

k=2: Zeta(s)*(1-1/2^(s-1))

k=3: Zeta(s)*(1-1/3^(s-1))

k=4: Zeta(s)*(1-1/2^(s-1))

k=5: Zeta(s)*(1-1/5^(s-1))

k=6: Zeta(s)*(1-1/2^(s-1)-1/3^(s-1)+1/6^(s-1))

k=7: Zeta(s)*(1-1/7^(s-1))

...

k=30: Zeta(s)*(1-1/2^(s-1)-1/3^(s-1)-1/5^(s-1)+1/6^(s-1)+1/10^(s-1)+1/15^(s-1)-1/30^(s-1))

...

(conjecture)

See triangle A142971 for negative distinct prime factors.

This could probably be checked by matrix multiplication.

The signs of the eigenvalues of this matrix are a rearrangement of the Mobius function A008683 (conjecture). The first few eigenvalues are:

{1.0000}

{-1.4142, 1.4142}

{-2.6554, 1.8662, -1.2108}

{-3.4393, 2.1004, -1.6611, 0}

{-4.7711, -3.3867, 2.5910, -1.4332, 0}

{-5.2439, -3.4641, 3.4641, 2.5169, -2.2730, 0}

The relation to Dirichlet characters for the entries in this matrix appears appears to be formulated in terms of the sequence A089026 which is equal to n if n is a prime, otherwise equal to 1. See Mathematica program below. [Mats Granvik, Nov 23 2013]

LINKS

Table of n, a(n) for n=1..91.

FORMULA

T(n,1)=1, T(1,k)=1, n>=k: -Sum from i=1 to k-1 of T(n-i,k), n<k: -Sum from i=1 to n-1 of T(k-i,n).

T(n, n) = A023900(n). - Michael Somos, Jul 18 2011

T(n,k) = A023900(GCD(n,k)). Conjecture: The Dirichlet generating function for the n-th row is Zeta[s] Total[1/Divisors[n]^(s - 1) * MoebiusMu[Divisors[n]]]]. - Mats Granvik, Jun 18 2012

EXAMPLE

Array starts:

1..1..1..1..1..1..1..1..1..1...

1.-1..1.-1..1.-1..1.-1..1.-1...

1..1.-2..1..1.-2..1..1.-2..1...

1.-1..1.-1..1.-1..1.-1..1.-1...

1..1..1..1.-4..1..1..1..1.-4...

1.-1.-2.-1..1..2..1.-1.-2.-1...

1..1..1..1..1..1.-6..1..1..1...

1.-1..1.-1..1.-1..1.-1..1.-1...

1..1.-2..1..1.-2..1..1.-2..1...

1.-1..1.-1.-4.-1..1.-1..1..4...

MATHEMATICA

T[ n_, k_] := T[ n, k] = If[ n < 1 || k < 1, 0, If[ n == 1 || k == 1, 1, If[ k > n, T[ k, n], If[ n > k, T[ k, Mod[ n, k, 1]], -Sum[ T[ n, i], {i, n - 1}]]]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] (* Michael Somos, Jul 18 2011 *)

nn = 12; b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn}]; j = 1; MatrixForm[Table[Table[Product[(b[[n]][[m]] * DirichletCharacter[b[[n]][[m]], j, k] - (b[[n]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]] (* Conjectured expression as Dirichlet characters. Mats Granvik, Nov 23 2013 *)

PROG

(PARI) {T(n, k) = if( n<1 || k<1, 0, if( n==1 || k==1, 1, if( k>n, T(k, n), if( k<n, T(k, (n-1)%k+1), -sum( i=1, n-1, T(n, i))))))} /* Michael Somos, Jul 18 2011 */

CROSSREFS

Cf. A023900, A014963, A008302, A033999, A061347, A142971.

Sequence in context: A113515 A103754 A058665 * A043290 A194333 A203640

Adjacent sequences:  A191895 A191896 A191897 * A191899 A191900 A191901

KEYWORD

sign,tabl

AUTHOR

Mats Granvik, Jun 19 2011

STATUS

approved

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Last modified October 26 02:56 EDT 2014. Contains 248566 sequences.