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A331573
The bottom entry in the forward difference table of the Euler totient function phi for 1..n.
1
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
1,4
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.
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
EXAMPLE
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.
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]]
CROSSREFS
KEYWORD
sign
AUTHOR
Robert G. Wilson v, Jan 20 2020
STATUS
approved