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

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, 260, 315, 384, 472, 584, 725, 900, 1114, 1372, 1679, 2041, 2466, 2965, 3553, 4250, 5082, 6081, 7285, 8738, 10490, 12597, 15121, 18130, 21699, 25912, 30865, 36670

Offset

1,5

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{m>=1} (x^(m^2) - x^(m*(m+1)))/(1-x)^m.

Example

a(9)=3 because we have [2,7], [7,2] and [3,3,3].

Maple

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
# second Maple program:
b:= proc(n, s, c) option remember; `if`(s<c, 0, `if`(n=0,
     `if`(s=c, 1, 0), add(b(n-j, min(j, s), c+1), j=1..n)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..52);  # Alois P. Heinz, Oct 01 2021

Mathematica

b[n_, s_, c_] := b[n, s, c] = If[s < c, 0, If[n == 0,
     If[s == c, 1, 0], Sum[b[n - j, Min[j, s], c + 1], {j, 1, n}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 1, 52}] (* Jean-François Alcover, Mar 13 2022, after Alois P. Heinz *)

Crossrefs

Cf. A006141.

Sequence in context: A352828 A029100 A356758 * A138185 A225941 A138705

Adjacent sequences: A098130 A098131 A098132 * A098134 A098135 A098136

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 27 2004

Extensions

More terms from Emeric Deutsch, Apr 18 2005

Status

approved