%I #17 Apr 14 2022 03:11:57

%S 0,1,3,6,13,28,59,122,248,501,1009,2028,4070,8159,16343,32717,65472,

%T 130991,262041,524157,1048410,2096943,4194043,8388285,16776819,

%U 33553946,67108270,134217002,268434568,536869825,1073740493,2147482019,4294965305,8589932164

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

%H Alois P. Heinz, <a href="/A348124/b348124.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) + A098132(n) + A098133(n) = 2^(n-1).

%p b:= proc(n, s, c) option remember; `if`(s<c, ceil(2^(n-1)),

%p `if`(n=0, 0, add(b(n-j, min(j, s), c+1), j=1..n)))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=1..40); # _Alois P. Heinz_, Oct 01 2021

%t b[n_, s_, c_] := b[n, s, c] = If[s < c, Ceiling[2^(n - 1)],

%t If[n == 0, 0, Sum[b[n - j, Min[j, s], c + 1], {j, 1, n}]]];

%t a[n_] := b[n, n, 0];

%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Apr 14 2022, after _Alois P. Heinz_ *)

%Y Cf. A011782, A098132, A098133.

%K nonn

%O 1,3

%A _R. J. Mathar_, Oct 01 2021

%E a(23)-a(34) from _Alois P. Heinz_, Oct 01 2021