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A317536 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} (1 + x^k)).

%I A317536 #9 Dec 16 2022 14:38:54
%S A317536 1,0,0,1,1,2,4,6,10,18,30,50,86,145,245,417,705,1193,2024,3427,5804,
%T A317536 9836,16660,28220,47811,80991,137197,232423,393729,666982,1129898,
%U A317536 1914078,3242495,5492898,9305130,15763154,26703273,45236138,76631348,129815818,219911870,372537244,631089250
%N A317536 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} (1 + x^k)).
%C A317536 Invert transform of A111133.
%H A317536 Alois P. Heinz, <a href="/A317536/b317536.txt">Table of n, a(n) for n = 0..4371</a>
%H A317536 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A317536 G.f.: 1/(1 - Sum_{k>=1} A111133(k)*x^k).
%p A317536 seq(coeff(series(1/(1+1/(1-x)-mul(1+x^k,k=1..n)), x,n+1),x,n),n=0..50); # _Muniru A Asiru_, Jul 30 2018
%p A317536 # second Maple program:
%p A317536 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A317536      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A317536     end:
%p A317536 a:= proc(n) option remember; `if`(n<1, 1,
%p A317536       add(a(n-i)*(b(i)-1), i=1..n))
%p A317536     end:
%p A317536 seq(a(n), n=0..42);  # _Alois P. Heinz_, Dec 16 2022
%t A317536 nmax = 42; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A317536 nmax = 42; CoefficientList[Series[1/(1 - Sum[(PartitionsQ[k] - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A317536 a[0] = 1; a[n_] := a[n] = Sum[(PartitionsQ[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]
%Y A317536 Cf. A000009, A050342, A089259, A111133, A270995, A279785, A299106, A304969, A305651, A317535.
%K A317536 nonn
%O A317536 0,6
%A A317536 _Ilya Gutkovskiy_, Jul 30 2018