A098504 Number of compositions of n such that every part occurs with the same multiplicity.

1, 1, 2, 4, 5, 6, 20, 14, 28, 49, 72, 66, 298, 134, 304, 646, 707, 618, 3794, 1178, 4856, 7926, 6300, 4758, 64004, 9267, 19624, 69346, 76148, 30462, 1491780, 55742, 294642, 1181578, 386820, 932804, 21400221, 315974, 1045372, 12081290, 66532116, 958266

Offset: 0

Author

Vladeta Jovovic, Oct 26 2004

Keywords

• nonn
• look

Examples

			a(6) = 20 because we have 6, 15, 51, 24, 42, 33, 123, 132, 213, 231, 312, 321, 222, 1122, 1212, 1221, 2112, 2121, 2211 and 111111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1300

Crossrefs

Cf. A047966, A032020.

Programs

Maple

G:= sum(sum((l*k)!/l!^k*x^(l*k*(k+1)/2)/product(1-x^(l*j),j=1..k), k=1..40),l=1..55):Gser:=series(G,x=0,55):seq(coeff(Gser,x^n), n=1..46); # Emeric Deutsch, Mar 28 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
       expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i-1)*x))))
    end:
a:= n-> `if`(n=0, 1, add((p-> add(coeff(p, x, i)*(i*m)!/(m!)^i,
        i=0..degree(p)))(b(n/m$2)), m=numtheory[divisors](n))):
seq(a(n), n=0..70);  # Alois P. Heinz, May 24 2014

Mathematica

b[n_, i_] := b[n, i] = If[n>i*(i+1)/2, 0, If[n == 0, 1, Expand[b[n, i-1] + If[i>n, 0, b[n-i, i-1]*x]]]]; a[n_] := If[n == 0, 1, Sum[Function[p, Sum[Coefficient[p, x, i]*(i*m)!/m!^i, {i, 0, Exponent[p, x]}]][b[n/m, n/m]], {m, Divisors[n]}]]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

Formula

G.f.: Sum(Sum((l*k)!/l!^k*x^(l*k*(k+1)/2)/Product(1-x^(l*j), j=1..k), k=1..infinity), l=1..infinity).

Extensions

More terms from Emeric Deutsch, Mar 28 2005

a(0)=1 from Alois P. Heinz, May 24 2014