cp's OEIS Frontend

A320650 Expansion of 1/(1 - Sum_{k>=1} x^k/(1 - x^(2*k))).

%I A320650 #13 Jun 16 2019 22:30:47
%S A320650 1,1,2,5,10,22,48,103,222,481,1038,2241,4842,10456,22582,48776,105342,
%T A320650 227514,491386,1061281,2292132,4950510,10692006,23092378,49874474,
%U A320650 107717891,232646956,502466304,1085216744,2343829586,5062156694,10933145610,23613191032
%N A320650 Expansion of 1/(1 - Sum_{k>=1} x^k/(1 - x^(2*k))).
%C A320650 Invert transform of A001227.
%H A320650 Robert Israel, <a href="/A320650/b320650.txt">Table of n, a(n) for n = 0..2986</a>
%H A320650 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A320650 G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^k)).
%F A320650 G.f.: 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1)))).
%F A320650 a(0) = 1; a(n) = Sum_{k=1..n} A001227(k)*a(n-k).
%p A320650 a:=series(1/(1-add(x^k/(1-x^(2*k)),k=1..100)),x=0,33): seq(coeff(a,x,n),n=0..32); # _Paolo P. Lava_, Apr 02 2019
%t A320650 nmax = 32; CoefficientList[Series[1/(1 - Sum[x^k/(1 - x^(2 k)), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A320650 nmax = 32; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A320650 a[0] = 1; a[n_] := a[n] = Sum[Sum[Mod[d, 2], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]
%Y A320650 Cf. A001227, A129921, A206303.
%K A320650 nonn
%O A320650 0,3
%A A320650 _Ilya Gutkovskiy_, Oct 18 2018