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Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).
5

%I #11 Jan 24 2024 10:09:51

%S 1,1,3,10,28,85,252,745,2202,6530,19326,57194,169341,501242,1483816,

%T 4392531,13002772,38491212,113943278,337298400,998482338,2955742400,

%U 8749688247,25901125616,76673399424,226971213462,671887935923,1988945626648,5887744768722,17429103155892,51594226501776

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

%C Invert transform of A022629.

%C a(n) is the number of compositions of n where there are A022629(k) sorts of part k. - _Joerg Arndt_, Jan 24 2024

%H G. C. Greubel, <a href="/A307063/b307063.txt">Table of n, a(n) for n = 0..1000</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} A022629(k)*a(n-k).

%t nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%o (Magma)

%o m:=80;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( 1/(2 - (&*[(1+j*x^j): j in [1..m+2]])) ));

%o (SageMath)

%o m=80;

%o def f(x): return 1/( 2 - product(1+j*x^j for j in range(1,m+3)) )

%o def A307063_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P( f(x) ).list()

%o A307063_list(m) # _G. C. Greubel_, Jan 24 2024

%Y Cf. A022629, A299164, A304969, A320652.

%Y Cf. A307057, A307058, A307059, A307060, A307062.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 21 2019