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A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).

%I A307058 #12 Jan 24 2024 10:08:52
%S A307058 1,1,1,2,4,7,12,21,38,68,120,212,377,670,1188,2107,3740,6638,11778,
%T A307058 20898,37084,65808,116775,207212,367696,652478,1157815,2054524,
%U A307058 3645730,6469316,11479734,20370656,36147506,64143372,113821732,201975429,358403220,635982680,1128544452,2002589998
%N A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).
%C A307058 Invert transform of A000700.
%H A307058 Alois P. Heinz, <a href="/A307058/b307058.txt">Table of n, a(n) for n = 0..2000</a>
%F A307058 a(0) = 1; a(n) = Sum_{k=1..n} A000700(k)*a(n-k).
%F A307058 From _G. C. Greubel_, Jan 24 2024: (Start)
%F A307058 G.f.: (1+x)/(2*(1+x) - x*QPochhammer(-1/x; x^2)).
%F A307058 G.f.: 1/( 2 - x^(1/24)*etx(x^2)^2/(eta(x^4)*eta(x)) ), where eta(x) is the Dedekind eta function. (End)
%p A307058 g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
%p A307058       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A307058     end:
%p A307058 a:= proc(n) option remember; `if`(n=0, 1,
%p A307058       add(a(n-i)*g(i), i=1..n))
%p A307058     end:
%p A307058 seq(a(n), n=0..39);  # _Alois P. Heinz_, Feb 09 2021
%t A307058 nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
%o A307058 (Magma)
%o A307058 m:=80;
%o A307058 R<x>:=PowerSeriesRing(Integers(), m);
%o A307058 Coefficients(R!( 1/(2 - (&*[1 + x^(2*j-1): j in [1..m+2]])) )); // _G. C. Greubel_, Jan 24 2024
%o A307058 (SageMath)
%o A307058 m=80;
%o A307058 def f(x): return 1/(2 - product(1+x^(2*j-1) for j in range(1,m+3)))
%o A307058 def A307058_list(prec):
%o A307058     P.<x> = PowerSeriesRing(QQ, prec)
%o A307058     return P( f(x) ).list()
%o A307058 A307058_list(m) # _G. C. Greubel_, Jan 24 2024
%Y A307058 Cf. A000700, A055887, A302017, A304969, A307059.
%Y A307058 Row sums of A341279.
%K A307058 nonn
%O A307058 0,4
%A A307058 _Ilya Gutkovskiy_, Mar 21 2019