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A336273 Number of compositions of n containing at least one part p of multiplicity p.

%I A336273 #11 Apr 14 2022 07:22:16
%S A336273 0,1,0,2,3,5,14,28,44,101,207,399,779,1609,3122,6121,11804,23631,
%T A336273 46273,91604,178096,352419,691996,1371306,2702206,5356324,10604748,
%U A336273 21080216,41869930,83383786,166114046,331434088,661685588,1322042390,2642367028,5283397304
%N A336273 Number of compositions of n containing at least one part p of multiplicity p.
%H A336273 Alois P. Heinz, <a href="/A336273/b336273.txt">Table of n, a(n) for n = 0..1000</a>
%F A336273 a(n) = A011782(n) - A336269(n).
%p A336273 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A336273       add(`if`(i=j, 0, b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
%p A336273     end:
%p A336273 a:= n-> ceil(2^(n-1))-b(n$2, 0):
%p A336273 seq(a(n), n=0..40);
%t A336273 b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
%t A336273      Sum[If[i == j, 0, b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
%t A336273 a[n_] := Ceiling[2^(n - 1)] - b[n, n, 0];
%t A336273 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 14 2022, after _Alois P. Heinz_ *)
%Y A336273 Cf. A011782, A336269.
%K A336273 nonn
%O A336273 0,4
%A A336273 _Alois P. Heinz_, Jul 15 2020