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A098133 Number of compositions of n in which the smallest part is equal to the number of parts.

%I A098133 #18 Mar 13 2022 09:50:23
%S A098133 1,0,0,1,2,2,2,2,3,5,8,11,14,17,20,24,30,39,52,69,90,115,144,177,215,
%T A098133 260,315,384,472,584,725,900,1114,1372,1679,2041,2466,2965,3553,4250,
%U A098133 5082,6081,7285,8738,10490,12597,15121,18130,21699,25912,30865,36670
%N A098133 Number of compositions of n in which the smallest part is equal to the number of parts.
%H A098133 Alois P. Heinz, <a href="/A098133/b098133.txt">Table of n, a(n) for n = 1..1000</a>
%F A098133 G.f.: Sum_{m>=1} (x^(m^2) - x^(m*(m+1)))/(1-x)^m.
%e A098133 a(9)=3 because we have [2,7], [7,2] and [3,3,3].
%p A098133 G:=sum((x^(m^2)-x^(m*(m+1)))/(1-x)^m,m=1..35):Gser:=series(G,x=0,60): seq(coeff(Gser,x^n),n=1..58); # _Emeric Deutsch_, Apr 18 2005
%p A098133 # second Maple program:
%p A098133 b:= proc(n, s, c) option remember; `if`(s<c, 0, `if`(n=0,
%p A098133      `if`(s=c, 1, 0), add(b(n-j, min(j, s), c+1), j=1..n)))
%p A098133     end:
%p A098133 a:= n-> b(n$2, 0):
%p A098133 seq(a(n), n=1..52);  # _Alois P. Heinz_, Oct 01 2021
%t A098133 b[n_, s_, c_] := b[n, s, c] = If[s < c, 0, If[n == 0,
%t A098133      If[s == c, 1, 0], Sum[b[n - j, Min[j, s], c + 1], {j, 1, n}]]];
%t A098133 a[n_] := b[n, n, 0];
%t A098133 Table[a[n], {n, 1, 52}] (* _Jean-François Alcover_, Mar 13 2022, after _Alois P. Heinz_ *)
%Y A098133 Cf. A006141.
%K A098133 easy,nonn
%O A098133 1,5
%A A098133 _Vladeta Jovovic_, Sep 27 2004
%E A098133 More terms from _Emeric Deutsch_, Apr 18 2005