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