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A321262 Expansion of 1/(1 - Sum_{k>=1} k*x^(2*k)/(1 - x^k)).

%I A321262 #7 Apr 02 2019 05:52:10
%S A321262 1,0,1,1,4,3,14,12,43,50,140,177,474,643,1560,2325,5246,8194,17763,
%T A321262 28838,60190,101063,204935,352227,700037,1224816,2394971,4250616,
%U A321262 8209174,14724570,28175997,50949079,96797183,176131780,332804667,608449008,1144920041,2100793404
%N A321262 Expansion of 1/(1 - Sum_{k>=1} k*x^(2*k)/(1 - x^k)).
%C A321262 Invert transform of A001065.
%H A321262 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A321262 G.f.: 1/(1 - Sum_{k>=1} (sigma(k) - k)*x^k).
%F A321262 G.f.: 1/(1 - Sum_{k>=1} (k - phi(k))*x^k/(1 - x^k)).
%F A321262 a(0) = 1; a(n) = Sum_{k=1..n} A001065(k)*a(n-k).
%p A321262 a:=series(1/(1-add(k*x^(2*k)/(1-x^k),k=1..100)),x=0,38): seq(coeff(a,x,n),n=0..37); # _Paolo P. Lava_, Apr 02 2019
%t A321262 nmax = 37; CoefficientList[Series[1/(1 - Sum[k x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A321262 nmax = 37; CoefficientList[Series[1/(1 - Sum[(k - EulerPhi[k]) x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A321262 a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[1, k] - k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 37}]
%Y A321262 Cf. A001065, A180305, A318784, A319107, A320651, A321263.
%K A321262 nonn
%O A321262 0,5
%A A321262 _Ilya Gutkovskiy_, Nov 01 2018