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A157159
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Infinite product representation of series 1 - log(1-x) = 1 + Sum_{j>=1} (j-1)!*(x^j)/j!.
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9
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1, 1, -1, 10, -16, 126, -526, 10312, -30024, 453840, -2805408, 45779328, -374664720, 7932770496, -67692115440, 2432120198016, -16610113920768, 437275706750208, -5110200130727808, 159305381515284480, -1931470594025607936, 63854116254680514048
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OFFSET
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1,4
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LINKS
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FORMULA
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Definition of a(n): 1-log(1-x) = product(1+a(n)*(x^n)/n!, n=1..infinity) (formal series and product).
Recurrence I. With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)) and the multinomial numbers M1(fp(n,m)) (given as array in A036038 for any partition) for fp(n,m) from FP(n,m): a(n) = (n-1)! - sum(sum(M1(fp)*product(a(k[j]),j=1..m),fp from FP(n,m)), m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=1, a(2)=1. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: a(n) = (n-1)!*((-1)^n + sum(d*(-a(d)/d!)^(n/d),d|n with 1<d<n)) + A089064(n), n>=2, a(1)=1. A089064(n)=sum(((-1)^(m-1))*(m-1)!)*|S1(n,m)|, m=1..n) with the unsigned Stirling numbers of the first kind |A008275|. See the W. Lang link under A147542 for these recurrences.
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EXAMPLE
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Recurrence I: a(7) = 6! - (7*a(1)*a(6) + 21*a(2)*a(5) + 35*a(3)*a(4) + 105*a(1)*a(2)*a(4)) = 720 - (7*126 + 21*(-16) + 35*(-1)*10 + 105*10) = -526.
Recurrence II: a(4) = 3!*(1+2*(-1/2!)^2) + 1 = +10.
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MAPLE
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a:= proc(n) option remember; `if`(n=1, 1, (n-1)!*((-1)^n+add(d*
(-a(d)/d!)^(n/d), d=numtheory[divisors](n) minus {1, n}))
+(-1)^(n+1)*add((k-1)!*Stirling1(n, k), k=1..n))
end:
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MATHEMATICA
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a[n_] := a[n] = If[n == 1, 1, (n-1)!*((-1)^n+Sum[d*(-a[d]/d!)^(n/d), {d, Divisors[n][[2 ;; -2]]}])+(-1)^(n+1)*Sum[(k-1)!*StirlingS1[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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