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A101561 a(n) = (-1)^n * [x^n] Sum_{k>=1} x^(k-1)/(1+3*x^k).

%I A101561 #18 Jun 25 2024 18:48:54
%S A101561 1,2,10,29,82,236,730,2216,6571,19604,59050,177410,531442,1593596,
%T A101561 4783060,14351123,43046722,129133838,387420490,1162281098,3486785140,
%U A101561 10460294156,31381059610,94143358424,282429536563,847288078004,2541865834900,7625599078610
%N A101561 a(n) = (-1)^n * [x^n] Sum_{k>=1} x^(k-1)/(1+3*x^k).
%H A101561 G. C. Greubel, <a href="/A101561/b101561.txt">Table of n, a(n) for n = 0..1000</a>
%F A101561 a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * A051731(n+1, k+1).
%F A101561 a(n) = (-1)^n * Sum_{d|n+1} (-3)^(d-1). - _G. C. Greubel_, Jun 25 2024
%t A101561 a[n_]:= Sum[(-1)^(n-k) * If[Mod[n+1, k+1]==0, 1, 0] * 3^k, {k, 0, n}];
%t A101561 Table[a[n], {n, 0, 25}] (* _James C. McMahon_, Jan 01 2024 *)
%t A101561 A101561[n_]:= (-1)^n*DivisorSum[n+1, (-3)^(#-1) &];
%t A101561 Table[A101561[n], {n,0,40}] (* _G. C. Greubel_, Jun 25 2024 *)
%o A101561 (Magma)
%o A101561 A101561:= func< n | (&+[(-1)^(n-k)*3^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
%o A101561 [A101561(n): n in [0..40]]; // _G. C. Greubel_, Jun 25 2024
%o A101561 (SageMath)
%o A101561 def A101561(n): return sum((-1)^(n+k)*3^k*0^((n+1)%(k+1)) for k in range(n+1))
%o A101561 [A101561(n) for n in range(41)] # _G. C. Greubel_, Jun 25 2024
%Y A101561 Cf. A048272, A051731, A081295, A101562, A101563.
%K A101561 easy,nonn
%O A101561 0,2
%A A101561 _Paul Barry_, Dec 07 2004