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

0,3

Links

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

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).

Example

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.

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 *)

Crossrefs

Cf. A047966, A032020.

Sequence in context: A336129 A008319 A033311 * A137653 A021411 A257433

Adjacent sequences: A098501 A098502 A098503 * A098505 A098506 A098507

Keyword

nonn,look

Author

Vladeta Jovovic, Oct 26 2004

Extensions

More terms from Emeric Deutsch, Mar 28 2005

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

Status

approved