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A098504
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Number of compositions of n such that every part occurs with the same multiplicity.
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35
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MAPLE
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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))):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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