# [seqfan] Re: A recurrence for the Dirichlet inverse of the Euler totient function.

Mats Granvik mgranvik at abo.fi
Sat Jun 18 18:13:23 CEST 2011

```I accidently replied only to Mitch so I repost here. The Mathematica program
for the Mahonian numbers is:

T[1, 1] = 1;
nn = 25;
T[n_, 1] = 0;
T[n_, k_] := T[n, k] = Sum[T[n - i, k - 1], {i, 1, k - 1}]
MatrixForm[Table[T[n, k], {n, nn}, {k, nn}]]

The recurrence for the Mahonian numbers takes values from the previous column
while the recurrence for array with the Dirichlet inverse of the Euler
totient function takes values from the same column or same row,
depending on which side of the main diagonal the element is.

Mats

Lainaus Mitch Harris <maharri at gmail.com>:

> Looking at the OEIS entries, I dont see the connection with the
> Mahonian numbers. Can you've more of how the recurrences are
> connected?
>
> Mitch
>
> On Jun 17, 2011, at 4:24 PM, "Mats Granvik" <mgranvik at abo.fi> wrote:
>
>> This Mathematica program generates the Dirichlet inverse of the Euler
>> totient function as the main diagonal of a symmetric array with
>> periodic columns and rows: https://oeis.org/A023900. The recurrence is
>> related to the recurrence for the Mahonian numbers:
>> https://oeis.org/A008302
>>
>> By changing signs in front of the sums we can get the
>> Fredholm-Rueppel sequence. https://oeis.org/A036987
>>
>> Mats Granvik
>>
>> Mathematica program:
>>
>> Clear[t, n, k, a, b];
>> nn = 50;
>> t[n_, 1] = 1;
>> t[1, k_] = 1;
>> t[n_, k_] :=
>>    t[n, k] =
>>     If[n < k,
>>      If[And[n > 1, k > 1], Sum[-t[k - i, n], {i, 1, n - 1}], 0],
>>      If[And[n > 1, k > 1], Sum[-t[n - i, k], {i, 1, k - 1}], 0]];
>> a = Flatten[Table[Table[t[n, k], {k, n, n}], {n, 1, nn}]];
>> a
>> Sign[a]
>> b = MoebiusMu[
>>     Prepend[Array[Times @@ First[Transpose[FactorInteger[#]]] &,
>>       nn - 1, 2], 1]];
>> Sign[a] - b
>>
>>
>>
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>>
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>

```