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

%I A098504 #33 Dec 21 2016 03:53:58
%S A098504 1,1,2,4,5,6,20,14,28,49,72,66,298,134,304,646,707,618,3794,1178,4856,
%T A098504 7926,6300,4758,64004,9267,19624,69346,76148,30462,1491780,55742,
%U A098504 294642,1181578,386820,932804,21400221,315974,1045372,12081290,66532116,958266
%N A098504 Number of compositions of n such that every part occurs with the same multiplicity.
%H A098504 Alois P. Heinz, <a href="/A098504/b098504.txt">Table of n, a(n) for n = 0..1300</a>
%F A098504 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).
%e A098504 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.
%p A098504 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
%p A098504 # second Maple program:
%p A098504 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
%p A098504        expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i-1)*x))))
%p A098504     end:
%p A098504 a:= n-> `if`(n=0, 1, add((p-> add(coeff(p, x, i)*(i*m)!/(m!)^i,
%p A098504         i=0..degree(p)))(b(n/m$2)), m=numtheory[divisors](n))):
%p A098504 seq(a(n), n=0..70);  # _Alois P. Heinz_, May 24 2014
%t A098504 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_ *)
%Y A098504 Cf. A047966, A032020.
%K A098504 nonn,look
%O A098504 0,3
%A A098504 _Vladeta Jovovic_, Oct 26 2004
%E A098504 More terms from _Emeric Deutsch_, Mar 28 2005
%E A098504 a(0)=1 from _Alois P. Heinz_, May 24 2014