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A052104
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Numerators of coefficients of the formal power series a(x) such that a(a(x)) = exp(x) - 1.
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4
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0, 1, 1, 1, 0, 1, -7, 1, 53, -281, -1231, 87379, -13303471, -54313201, 10142361989, 2821265977, -10502027401553, 1836446156249, 2952828271088741, -1004826382596003137, -7006246797736924249, 14607119841651449406947, 1868869263315549659372569
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OFFSET
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0,7
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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) = numerator(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. - Vladimir Kruchinin, Nov 08 2011
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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, m) T(n, m):= `if`(n=m, 1, (Stirling2(n, m)*m!/n!-
add(T(n, i)*T(i, m), i=m+1..n-1))/2)
end:
a:= n-> numer(T(n, 1)):
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MATHEMATICA
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T[n_, n_] = 1; T[n_, m_] := T[n, m] = (StirlingS2[n, m]*m!/n! - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])/2; Table[T[n, 1] // Numerator, {n, 0, 30}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
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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
[numerator(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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sign,nice,easy,frac
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AUTHOR
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STATUS
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approved
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