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A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).

%I A307062 #12 Jan 24 2024 09:56:15
%S A307062 1,1,3,10,29,88,264,790,2366,7086,21216,63523,190201,569485,1705121,
%T A307062 5105383,15286247,45769238,137039743,410316854,1228548190,3678451550,
%U A307062 11013817655,32976968175,98737827756,295635383297,885175234817,2650343093602,7935511791620,23760073760720,71141108467679
%N A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).
%C A307062 Invert transform of A026007.
%C A307062 a(n) is the number of compositions of n where there are A026007(k) sorts of part k. - _Joerg Arndt_, Jan 24 2024
%H A307062 G. C. Greubel, <a href="/A307062/b307062.txt">Table of n, a(n) for n = 0..1000</a>
%F A307062 a(0) = 1; a(n) = Sum_{k=1..n} A026007(k)*a(n-k).
%p A307062 b:= proc(n) b(n):= add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n)) end:
%p A307062 g:= proc(n) g(n):= `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n) end:
%p A307062 a:= proc(n) a(n):= `if`(n=0, 1, add(g(k)*a(n-k), k=1..n)) end:
%p A307062 seq(a(n), n=0..45);  # _Alois P. Heinz_, Jan 24 2024
%t A307062 nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%o A307062 (Magma)
%o A307062 m:=80;
%o A307062 R<x>:=PowerSeriesRing(Integers(), m);
%o A307062 Coefficients(R!( 1/(2 - (&*[(1+x^j)^j: j in [1..m+2]])) )); // _G. C. Greubel_, Jan 24 2024
%o A307062 (SageMath)
%o A307062 m=80;
%o A307062 def f(x): return 1/( 2 - product((1+x^j)^j for j in range(1,m+3)) )
%o A307062 def A307062_list(prec):
%o A307062     P.<x> = PowerSeriesRing(QQ, prec)
%o A307062     return P( f(x) ).list()
%o A307062 A307062_list(m) # _G. C. Greubel_, Jan 24 2024
%Y A307062 Cf. A026007, A257674, A299167, A304969.
%Y A307062 Cf. A307057, A307058, A307059, A307060, A307063.
%K A307062 nonn
%O A307062 0,3
%A A307062 _Ilya Gutkovskiy_, Mar 21 2019