cp's OEIS Frontend

A081295 a(n) = (-1)^(n+1) * coefficient of x^n in Sum_{k>=1} x^k/(1+2*x^k).

%I A081295 #15 Jun 22 2024 06:18:42
%S A081295 1,1,5,9,17,29,65,137,261,497,1025,2085,4097,8129,16405,32905,65537,
%T A081295 130845,262145,524793,1048645,2096129,4194305,8390821,16777233,
%U A081295 33550337,67109125,134225865,268435457,536855053,1073741825,2147516553,4294968325,8589869057
%N A081295 a(n) = (-1)^(n+1) * coefficient of x^n in Sum_{k>=1} x^k/(1+2*x^k).
%H A081295 Robert Israel, <a href="/A081295/b081295.txt">Table of n, a(n) for n = 1..3319</a>
%F A081295 a(n) = (-1)^(n+1) * [x^n]( Sum_{k>=1} x^k/(1+2*x^k) ).
%F A081295 a(p) = 2^(p-1) - 1, for p prime.
%F A081295 a(n) = (-1)^(n+1) * Sum_{d|n} (-2)^(d-1). - _Robert Israel_, Jun 04 2018
%F A081295 a(n) = (-1)^(n-1)*Sum_{k=1..n} (-1)^(k-1)*A128315(n, k). - _G. C. Greubel_, Jun 22 2024
%p A081295 f:= n -> (-1)^(n+1)*add((-2)^(d-1),d=numtheory:-divisors(n)):
%p A081295 map(f, [$1..100]); # _Robert Israel_, Jun 04 2018
%t A081295 A081295[n_]:= (-1)^(n+1)*DivisorSum[n, (-2)^(#-1) &];
%t A081295 Table[A081295[n], {n, 40}] (* _G. C. Greubel_, Jun 22 2024 *)
%o A081295 (PARI) a(n) =if(n<1, 0, (-1)^(n+1)*polcoeff(sum(k=1, n, x^k/(1+2*x^k), x*O(x^n)), n))
%o A081295 (Magma)
%o A081295 A081295:= func< n | (-1)^(n+1)*(&+[(-2)^(d-1): d in Divisors(n)]) >;
%o A081295 [A081295(n): n in [1..40]]; // _G. C. Greubel_, Jun 22 2024
%o A081295 (SageMath)
%o A081295 def A081295(n): return (-1)^(n+1)*sum((-2)^(k-1) for k in (1..n) if (k).divides(n))
%o A081295 [A081295(n) for n in range(1,41)] # _G. C. Greubel_, Jun 22 2024
%Y A081295 Cf. A034729, A128315.
%K A081295 nonn
%O A081295 1,3
%A A081295 _Benoit Cloitre_, Apr 20 2003