cp's OEIS Frontend

A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.

%I A382993 #19 Apr 12 2025 11:21:34
%S A382993 1,2,0,3,-1,1,4,-3,4,0,5,-6,11,-4,1,6,-10,24,-21,8,0,7,-15,45,-66,51,
%T A382993 -10,1,8,-21,76,-160,208,-119,20,0,9,-28,119,-330,629,-676,315,-34,1,
%U A382993 10,-36,176,-609,1560,-2590,2344,-831,60,0,11,-45,249,-1036,3367,-7750,11165,-8226,2195,-100,1
%N A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.
%F A382993 A(n,k) = (1/n) * A382994(n,k).
%F A382993 A(n,k) = -(1/n) * Sum_{j=1..n} (-k)^gcd(n,j).
%F A382993 G.f. of column k: Sum_{j>=1} phi(j) * log(1 + k*x^j) / j.
%F A382993 Product_{n>=1} 1/(1 - x^n)^A(n,k) = Product_{n>=1} (1 + k*x^n).
%e A382993 Square array begins:
%e A382993   1,   2,    3,    4,     5,     6,      7, ...
%e A382993   0,  -1,   -3,   -6,   -10,   -15,    -21, ...
%e A382993   1,   4,   11,   24,    45,    76,    119, ...
%e A382993   0,  -4,  -21,  -66,  -160,  -330,   -609, ...
%e A382993   1,   8,   51,  208,   629,  1560,   3367, ...
%e A382993   0, -10, -119, -676, -2590, -7750, -19565, ...
%e A382993   1,  20,  315, 2344, 11165, 39996, 117655, ...
%o A382993 (PARI) a(n, k) = -sumdiv(n, d, eulerphi(n/d)*(-k)^d)/n;
%Y A382993 Columns k=1..5 give A000035, (-1)^(n+1) * A074763(n), A343465, A343466, A343467.
%Y A382993 Main diagonal gives A382998.
%Y A382993 Cf. A000010, A075195, A286957, A382994, A383011.
%K A382993 sign,tabl
%O A382993 1,2
%A A382993 _Seiichi Manyama_, Apr 11 2025