cp's OEIS Frontend

A344820 a(n) = Sum_{k=1..n} floor(n/k) * (-n)^(k-1).

%I A344820 #25 Jun 27 2024 01:38:34
%S A344820 1,0,9,-52,520,-6636,102984,-1864600,38741463,-909081740,23775986069,
%T A344820 -685854111804,21633935838489,-740800448012044,27368368159530285,
%U A344820 -1085102592823737200,45957792326631241516,-2070863582899905915336,98920982783031811482920,-4993219047619535240997780
%N A344820 a(n) = Sum_{k=1..n} floor(n/k) * (-n)^(k-1).
%H A344820 Seiichi Manyama, <a href="/A344820/b344820.txt">Table of n, a(n) for n = 1..387</a>
%F A344820 a(n) = Sum_{k=1..n} Sum_{d|k} (-n)^(d-1).
%F A344820 a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k/(1 + n*x^k).
%F A344820 a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (-n)^(k-1) * x^k/(1 - x^k).
%F A344820 a(n) ~ -(-1)^n * n^(n-1). - _Vaclav Kotesovec_, Jun 05 2021
%t A344820 a[n_] := Sum[(-n)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 20] (* _Amiram Eldar_, May 29 2021 *)
%o A344820 (PARI) a(n) = sum(k=1, n, n\k*(-n)^(k-1));
%o A344820 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-n)^(d-1)));
%o A344820 (Magma)
%o A344820 A344820:= func< n | (&+[Floor(n/k)*(-n)^(k-1): k in [1..n]]) >;
%o A344820 [A344820(n): n in [1..40]]; // _G. C. Greubel_, Jun 26 2024
%o A344820 (SageMath)
%o A344820 def A344820(n): return sum((n//k)*(-n)^(k-1) for k in range(1,n+1))
%o A344820 [A344820(n) for n in range(1,41)] # _G. C. Greubel_, Jun 26 2024
%Y A344820 Diagonal of A344824.
%Y A344820 Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): this sequence (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).
%K A344820 sign
%O A344820 1,3
%A A344820 _Seiichi Manyama_, May 29 2021