cp's OEIS Frontend

A382995 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = Sum_{d|n} phi(n/d) * (-k)^(d-1).

%I A382995 #15 Apr 12 2025 09:49:37
%S A382995 1,1,0,1,-1,3,1,-2,6,0,1,-3,11,-8,5,1,-4,18,-28,20,0,1,-5,27,-66,85,
%T A382995 -30,7,1,-6,38,-128,260,-238,70,0,1,-7,51,-220,629,-1014,735,-136,9,1,
%U A382995 -8,66,-348,1300,-3108,4102,-2216,270,0,1,-9,83,-518,2405,-7750,15631,-16452,6585,-500,11
%N A382995 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = Sum_{d|n} phi(n/d) * (-k)^(d-1).
%F A382995 A(n,k) = (1/k) * A382994(n,k).
%F A382995 A(n,k) = Sum_{j=1..n} (-k)^(gcd(n,j) - 1).
%F A382995 G.f. of column k: Sum_{j>=1} phi(j) * x^j / (1 + k*x^j).
%e A382995 Square array begins:
%e A382995   1,   1,    1,     1,     1,     1,      1, ...
%e A382995   0,  -1,   -2,    -3,    -4,    -5,     -6, ...
%e A382995   3,   6,   11,    18,    27,    38,     51, ...
%e A382995   0,  -8,  -28,   -66,  -128,  -220,   -348, ...
%e A382995   5,  20,   85,   260,   629,  1300,   2405, ...
%e A382995   0, -30, -238, -1014, -3108, -7750, -16770, ...
%e A382995   7,  70,  735,  4102, 15631, 46662, 117655, ...
%o A382995 (PARI) a(n, k) = sumdiv(n, d, eulerphi(n/d)*(-k)^(d-1));
%Y A382995 Columns k=1..3 give A193356, A382999, A383000.
%Y A382995 Main diagonal gives A382998.
%Y A382995 Cf. A000010, A343489, A382994.
%K A382995 sign,tabl
%O A382995 1,6
%A A382995 _Seiichi Manyama_, Apr 12 2025