cp's OEIS Frontend

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

%I A331573 #25 Oct 03 2022 04:46:20
%S A331573 1,0,1,-2,5,-14,39,-102,247,-558,1197,-2494,5167,-10850,23311,-51132,
%T A331573 113333,-250694,547871,-1175998,2475153,-5117486,10439895,-21142030,
%U A331573 42777735,-86960284,178221401,-368541508,767762191,-1606535062,3365499467,-7038925364,14671422797,-30450115592
%N A331573 The bottom entry in the forward difference table of the Euler totient function phi for 1..n.
%C A331573 a(2n) is a nonpositive even number while a(2n-1) is an odd positive number.
%C A331573 Abs(a(n)) < abs(a(n+1)) for 1 < n < 8000.
%F A331573 a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*phi(k). - _Ridouane Oudra_, Aug 21 2021
%F A331573 a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*A002088(k). - _Ridouane Oudra_, Oct 02 2022
%e A331573 a(8) = -102 because:
%e A331573 1     1     2     2     4     2     6     4  (first 8 terms of A000010)
%e A331573    0     1     0     2    -2     4    -2     (first 7 terms of A057000)
%e A331573       1    -1     2    -4     6     6
%e A331573         -2     3    -6    10   -12
%e A331573             5    -9    16   -22
%e A331573              -14    25   -38
%e A331573                  39   -63
%e A331573                   -102
%e A331573 The first principal right descending diagonal is this sequence.
%t A331573 f[n_] := Differences[ Array[ EulerPhi, n], n -1][[1]]; Array[f, 34] (* or *)
%t A331573 nmx = 34; Join[ {1}, Differences[ Array[ EulerPhi, nmx], #][[1]] & /@ Range[nmx - 1]]
%Y A331573 Cf. A187202, A000010, A057000.
%Y A331573 Cf. A002088.
%K A331573 sign
%O A331573 1,4
%A A331573 _Robert G. Wilson v_, Jan 20 2020