A331573 The bottom entry in the forward difference table of the Euler totient function phi for 1..n.

1, 0, 1, -2, 5, -14, 39, -102, 247, -558, 1197, -2494, 5167, -10850, 23311, -51132, 113333, -250694, 547871, -1175998, 2475153, -5117486, 10439895, -21142030, 42777735, -86960284, 178221401, -368541508, 767762191, -1606535062, 3365499467, -7038925364, 14671422797, -30450115592

Offset: 1

Author

Robert G. Wilson v, Jan 20 2020

Keywords

• sign

Comments

a(2n) is a nonpositive even number while a(2n-1) is an odd positive number.

Abs(a(n)) < abs(a(n+1)) for 1 < n < 8000.

Examples

			a(8) = -102 because:
1     1     2     2     4     2     6     4  (first 8 terms of A000010)
   0     1     0     2    -2     4    -2     (first 7 terms of A057000)
      1    -1     2    -4     6     6
        -2     3    -6    10   -12
            5    -9    16   -22
             -14    25   -38
                 39   -63
                  -102
The first principal right descending diagonal is this sequence.
		

Crossrefs

Cf. A187202, A000010, A057000.

Cf. A002088.

Programs

Mathematica

f[n_] := Differences[ Array[ EulerPhi, n], n -1][[1]]; Array[f, 34] (* or *)
nmx = 34; Join[ {1}, Differences[ Array[ EulerPhi, nmx], #][[1]] & /@ Range[nmx - 1]]

Formula

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*phi(k). - Ridouane Oudra, Aug 21 2021

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*A002088(k). - Ridouane Oudra, Oct 02 2022