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%I A341020 #13 Apr 14 2022 03:11:54
%S A341020 1,1,2,5,10,22,48,104,224,485,1050,2271,4914,10633,23006,49777,107700,
%T A341020 233027,504192,1090901,2360344,5106990,11049806,23908060,51728994,
%U A341020 111924129,242166136,523965997,1133686030,2452914922,5307282140,11483171904,24845718294
%N A341020 INVERT transform of the binary weight.
%C A341020 Number of compositions of n into parts where there are A000120(k) sorts of part k.
%H A341020 Alois P. Heinz, <a href="/A341020/b341020.txt">Table of n, a(n) for n = 0..2984</a>
%F A341020 G.f.: 1 / (1 - Sum_{k>=1} A000120(k) * x^k).
%p A341020 h:= proc(n) option remember; add(j, j=Bits[Split](n)) end:
%p A341020 a:= proc(n) option remember; `if`(n=0, 1,
%p A341020       add(a(n-i)*h(i), i=1..n))
%p A341020     end:
%p A341020 seq(a(n), n=0..36);
%t A341020 h[n_] := h[n] = DigitCount[n, 2, 1];
%t A341020 a[n_] := a[n] = If[n == 0, 1,
%t A341020      Sum[a[n - i]*h[i], {i, 1, n}]];
%t A341020 Table[a[n], {n, 0, 36}] (* _Jean-François Alcover_, Apr 14 2022, after _Alois P. Heinz_ *)
%Y A341020 Cf. A000120.
%K A341020 nonn
%O A341020 0,3
%A A341020 _Alois P. Heinz_, Feb 02 2021

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