A320650 - OEIS

%I #13 Jun 16 2019 22:30:47

%S 1,1,2,5,10,22,48,103,222,481,1038,2241,4842,10456,22582,48776,105342,

%T 227514,491386,1061281,2292132,4950510,10692006,23092378,49874474,

%U 107717891,232646956,502466304,1085216744,2343829586,5062156694,10933145610,23613191032

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

%C Invert transform of A001227.

%H Robert Israel, <a href="/A320650/b320650.txt">Table of n, a(n) for n = 0..2986</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^k)).

%F G.f.: 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1)))).

%F a(0) = 1; a(n) = Sum_{k=1..n} A001227(k)*a(n-k).

%p 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 nmax = 32; CoefficientList[Series[1/(1 - Sum[x^k/(1 - x^(2 k)), {k, 1, nmax}]), {x, 0, nmax}], x]

%t nmax = 32; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%t 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 Cf. A001227, A129921, A206303.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Oct 18 2018