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a(n) = Sum_{k=1..n} (-1)^k*k^n*floor(n/k).
2

%I #13 Oct 29 2023 22:05:59

%S -1,2,-22,203,-2285,33855,-609345,12420372,-284964519,7347342215,

%T -209807114169,6554034238459,-222469737401739,8159109186320903,

%U -321461264348047819,13538455640979049698,-606976994365011212414,28864017965496692865925,-1451086990386146504580735

%N a(n) = Sum_{k=1..n} (-1)^k*k^n*floor(n/k).

%F a(n) = (-1)^n*A308313(n).

%F Let A(n,k) = Sum_{j=1..n} j^k * floor(n/j). Then a(n) = 2^(n+1)*A(floor(n/2),n)-A(n,n).

%t a[n_]:=Sum[ (-1)^k*k^n*Floor[n/k],{k,n}]; Array[a,19] (* _Stefano Spezia_, Oct 29 2023 *)

%o (Python)

%o from math import isqrt

%o from sympy import bernoulli

%o def A366919(n): return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))<<n+1)-((t:=isqrt(n))+1)*(bernoulli(n+1)-bernoulli(n+1,t+1))+(sum(w**n*(n+1)*((q:=m//w)+1)-bernoulli(n+1)+bernoulli(n+1,q+1) for w in range(1,s+1))<<n+1)-sum(w**n*(n+1)*((q:=n//w)+1)-bernoulli(n+1)+bernoulli(n+1,q+1) for w in range(1,t+1)))//(n+1)

%o (PARI) a(n) = sum(k=1, n, (-1)^k*k^n*(n\k)); \\ _Michel Marcus_, Oct 29 2023

%Y Cf. A024919, A308313, A366915, A366917, A319649.

%K sign

%O 1,2

%A _Chai Wah Wu_, Oct 28 2023