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A348124 Number of compositions of n where the smallest part is smaller than the number of parts.

%I A348124 #17 Apr 14 2022 03:11:57
%S A348124 0,1,3,6,13,28,59,122,248,501,1009,2028,4070,8159,16343,32717,65472,
%T A348124 130991,262041,524157,1048410,2096943,4194043,8388285,16776819,
%U A348124 33553946,67108270,134217002,268434568,536869825,1073740493,2147482019,4294965305,8589932164
%N A348124 Number of compositions of n where the smallest part is smaller than the number of parts.
%H A348124 Alois P. Heinz, <a href="/A348124/b348124.txt">Table of n, a(n) for n = 1..1000</a>
%F A348124 a(n) + A098132(n) + A098133(n) = 2^(n-1).
%p A348124 b:= proc(n, s, c) option remember; `if`(s<c, ceil(2^(n-1)),
%p A348124       `if`(n=0, 0, add(b(n-j, min(j, s), c+1), j=1..n)))
%p A348124     end:
%p A348124 a:= n-> b(n$2, 0):
%p A348124 seq(a(n), n=1..40);  # _Alois P. Heinz_, Oct 01 2021
%t A348124 b[n_, s_, c_] := b[n, s, c] = If[s < c, Ceiling[2^(n - 1)],
%t A348124      If[n == 0, 0, Sum[b[n - j, Min[j, s], c + 1], {j, 1, n}]]];
%t A348124 a[n_] := b[n, n, 0];
%t A348124 Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Apr 14 2022, after _Alois P. Heinz_ *)
%Y A348124 Cf. A011782, A098132, A098133.
%K A348124 nonn
%O A348124 1,3
%A A348124 _R. J. Mathar_, Oct 01 2021
%E A348124 a(23)-a(34) from _Alois P. Heinz_, Oct 01 2021