cp's OEIS Frontend

A382994 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.

%I A382994 #15 Apr 12 2025 11:21:18
%S A382994 1,2,0,3,-2,3,4,-6,12,0,5,-12,33,-16,5,6,-20,72,-84,40,0,7,-30,135,
%T A382994 -264,255,-60,7,8,-42,228,-640,1040,-714,140,0,9,-56,357,-1320,3145,
%U A382994 -4056,2205,-272,9,10,-72,528,-2436,7800,-15540,16408,-6648,540,0
%N A382994 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.
%F A382994 A(n,k) = -Sum_{j=1..n} (-k)^gcd(n,j).
%F A382994 G.f. of column k: k * Sum_{j>=1} phi(j) * x^j / (1 + k*x^j).
%e A382994 Square array begins:
%e A382994   1,   2,    3,     4,      5,      6,       7, ...
%e A382994   0,  -2,   -6,   -12,    -20,    -30,     -42, ...
%e A382994   3,  12,   33,    72,    135,    228,     357, ...
%e A382994   0, -16,  -84,  -264,   -640,  -1320,   -2436, ...
%e A382994   5,  40,  255,  1040,   3145,   7800,   16835, ...
%e A382994   0, -60, -714, -4056, -15540, -46500, -117390, ...
%e A382994   7, 140, 2205, 16408,  78155, 279972,  823585, ...
%o A382994 (PARI) a(n, k) = -sumdiv(n, d, eulerphi(n/d)*(-k)^d);
%Y A382994 Main diagonal gives A382997.
%Y A382994 Cf. A000010, A185651, A382993, A382995.
%K A382994 sign,tabl
%O A382994 1,2
%A A382994 _Seiichi Manyama_, Apr 12 2025