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A159929 INVERT transform of phi(n), A000010.

%I A159929 #25 Aug 18 2021 12:35:16
%S A159929 1,1,2,5,11,26,57,131,296,669,1515,3430,7765,17577,39790,90069,203897,
%T A159929 461562,1044847,2365239,5354224,12120455,27437267,62110208,140599921,
%U A159929 318278385,720492104,1630990029,3692099407,8357867190,18919843773,42829166807,96953101328,219474357191,496827773575
%N A159929 INVERT transform of phi(n), A000010.
%C A159929 Number of compositions of n into parts where there are phi(k) sorts of part k. - _Joerg Arndt_, Sep 30 2012
%H A159929 Alois P. Heinz, <a href="/A159929/b159929.txt">Table of n, a(n) for n = 0..1000</a>
%F A159929 INVERT transform of A000010.
%F A159929 G.f.: 1/( 1 - Sum_{k>=1} phi(k) * x^k ) where phi = A000010. _Joerg Arndt_, Sep 30 2012
%F A159929 a(n) ~ c * d^n, where d = 2.26371672715382105671101924573765243871241560288177676216035633730282369149... is the root of the equation Sum_{k>=1} phi(k)/d^k = 1 and c = 0.42880036544961338799475947921442516792321060146527623589359809145075482942... - _Vaclav Kotesovec_, Aug 18 2021
%e A159929 a(6) = 57 = (1, 1, 2, 2, 4, 2) dot (26, 11, 5, 2, 1, 1) = (26 + 11 + 10 + 4 + 4 + 2).
%p A159929 a:= proc(n) option remember; `if`(n=0, 1,
%p A159929       add(a(n-i)*numtheory[phi](i), i=1..n))
%p A159929     end:
%p A159929 seq(a(n), n=0..35);  # _Alois P. Heinz_, Sep 22 2017
%t A159929 a[n_] := a[n] = If[n == 0, 1, Sum[a[n-i] EulerPhi[i], {i, 1, n}]];
%t A159929 a /@ Range[0, 35] (* _Jean-François Alcover_, Oct 31 2020, after Maple *)
%o A159929 (PARI)
%o A159929 N=66;  x='x+O('x^N);
%o A159929 Vec( 1/( 1 - sum(k=1,N, eulerphi(k)*x^k ) ) - 1 )
%o A159929 /* _Joerg Arndt_, Sep 30 2012 */
%Y A159929 Cf. A000010.
%Y A159929 Row sums of A340995.
%K A159929 nonn
%O A159929 0,3
%A A159929 _Gary W. Adamson_, Apr 26 2009
%E A159929 a(0)=1 prepended by _Alois P. Heinz_, Sep 22 2017