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A052105
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Denominators of coefficients in the formal power series a(x) such that a(a(x)) = exp(x) - 1.
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4
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1, 1, 4, 48, 1, 3840, 92160, 645120, 3440640, 30965760, 14863564800, 24222105600, 7847962214400, 40809403514880, 5713316492083200, 7617755322777600, 5484783832399872000, 5328075722902732800, 1220613711064989696000
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OFFSET
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0,3
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.52c.
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LINKS
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FORMULA
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a(n) = denominator(T(n,1)) where T(n, m) = if n=m then 1, otherwise ( StirlingS2(n, m)*m!/n! - Sum_{i=m+1..n-1} T(n, i) * T(i, m)))/2. - G. C. Greubel, Apr 15 2021
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EXAMPLE
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a(x) = x + x^2/4 + x^3/48 + x^5/3840 - 7*x^6/92160 + x^7/645120 + ...
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MAPLE
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T:= proc(n, k);
T(n, k):= `if`(n=k, 1, (Stirling2(n, k)*k!/n! - add(T(n, j)*T(j, k), j = k+1..n-1))/2);
end proc;
a:= n -> denom(T(n, 1));
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MATHEMATICA
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(* First program *)
a[x_, n_] := Sum[c[k] x^k, {k, 0, n}] ;
f[x_, n_] := Series[Exp[x] - 1, {x, 0, n}] // Normal;
b[x_, n_] := Series[a[a[x, n], n], {x, 0, n}] // Normal;
eq[n_] := Thread[CoefficientList[f[x, n] - b[x, n], x] == 0] // Rest;
c[0] = 0; so[3] = Solve[eq[3], {c[1], c[2], c[3]}] // First;
so[n_] := so[n] = Solve[eq[n] /. Flatten[Table[so[k], {k, 3, n - 1}]], c[n]] // First
Array[c, 19, 0] /. Flatten[Table[so[k], {k, 3, 19}]] // Denominator
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==n, 1, (StirlingS2[n, k]*k!/n! - Sum[T[n, j]*T[j, k], {j, k+1, n-1}])/2];
Table[Denominator[T[n, 1]], {n, 0, 30}] (* G. C. Greubel, Apr 15 2021 *)
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PROG
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(Sage)
@CachedFunction
def T(n, k):
if (k==n): return 1
else: return ( (factorial(k)/factorial(n))*stirling_number2(n, k) - sum(T(n, j)*T(j, k) for j in (k+1..n-1)) )/2
[denominator(T(n, 1)) for n in (0..30)] # G. C. Greubel, Apr 15 2021
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CROSSREFS
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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STATUS
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approved
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