cp's OEIS Frontend

A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).

%I A307063 #11 Jan 24 2024 10:09:51
%S A307063 1,1,3,10,28,85,252,745,2202,6530,19326,57194,169341,501242,1483816,
%T A307063 4392531,13002772,38491212,113943278,337298400,998482338,2955742400,
%U A307063 8749688247,25901125616,76673399424,226971213462,671887935923,1988945626648,5887744768722,17429103155892,51594226501776
%N A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).
%C A307063 Invert transform of A022629.
%C A307063 a(n) is the number of compositions of n where there are A022629(k) sorts of part k. - _Joerg Arndt_, Jan 24 2024
%H A307063 G. C. Greubel, <a href="/A307063/b307063.txt">Table of n, a(n) for n = 0..1000</a>
%F A307063 a(0) = 1; a(n) = Sum_{k=1..n} A022629(k)*a(n-k).
%t A307063 nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%o A307063 (Magma)
%o A307063 m:=80;
%o A307063 R<x>:=PowerSeriesRing(Integers(), m);
%o A307063 Coefficients(R!( 1/(2 - (&*[(1+j*x^j): j in [1..m+2]])) ));
%o A307063 (SageMath)
%o A307063 m=80;
%o A307063 def f(x): return 1/( 2 - product(1+j*x^j for j in range(1,m+3)) )
%o A307063 def A307063_list(prec):
%o A307063     P.<x> = PowerSeriesRing(QQ, prec)
%o A307063     return P( f(x) ).list()
%o A307063 A307063_list(m) # _G. C. Greubel_, Jan 24 2024
%Y A307063 Cf. A022629, A299164, A304969, A320652.
%Y A307063 Cf. A307057, A307058, A307059, A307060, A307062.
%K A307063 nonn
%O A307063 0,3
%A A307063 _Ilya Gutkovskiy_, Mar 21 2019