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A157159 Infinite product representation of series 1 - log(1-x)= 1 + sum((j-1)!*(x^j)/j!, j=1..infinity). 2
1, 1, -1, 10, -16, 126, -526, 10312, -30024, 453840, -2805408, 45779328, -374664720, 7932770496, -67692115440, 2432120198016, -16610113920768, 437275706750208, -5110200130727808, 159305381515284480, -1931470594025607936, 63854116254680514048 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..170

FORMULA

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.

EXAMPLE

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.

MAPLE

with (numtheory): with (combinat):

a:= proc(n) option remember; `if`(n=1, 1,

      (n-1)!*((-1)^n +add(d*(-a(d)/d!)^(n/d), d=divisors(n) minus {1, n}))

      +(-1)^(n+1)*add((k-1)!*stirling1(n, k), k=1..n))

    end:

seq (a(n), n=1..30);  # Alois P. Heinz, Aug 14 2012

MATHEMATICA

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-Fran├žois Alcover, Mar 05 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A137852, A006973, A147542.

Sequence in context: A219854 A167331 A255531 * A176664 A256346 A079630

Adjacent sequences:  A157156 A157157 A157158 * A157160 A157161 A157162

KEYWORD

sign

AUTHOR

Wolfdieter Lang Mar 06 2009

EXTENSIONS

More terms from Alois P. Heinz, Aug 14 2012

STATUS

approved

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Last modified July 5 23:36 EDT 2020. Contains 335475 sequences. (Running on oeis4.)