# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a157159
Showing 1-1 of 1

%I A157159 #15 Jun 03 2022 09:43:22
%S A157159 1,1,-1,10,-16,126,-526,10312,-30024,453840,-2805408,45779328,
%T A157159 -374664720,7932770496,-67692115440,2432120198016,-16610113920768,
%U A157159 437275706750208,-5110200130727808,159305381515284480,-1931470594025607936,63854116254680514048
%N A157159 Infinite product representation of series 1 - log(1-x) = 1 + Sum_{j>=1} (j-1)!*(x^j)/j!.
%H A157159 Alois P. Heinz, <a href="/A157159/b157159.txt">Table of n, a(n) for n = 1..170</a>
%F A157159 Definition of a(n): 1-log(1-x) = product(1+a(n)*(x^n)/n!, n=1..infinity) (formal series and product).
%F A157159 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).
%F A157159 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.
%e A157159 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.
%e A157159 Recurrence II: a(4) = 3!*(1+2*(-1/2!)^2) + 1 = +10.
%p A157159 a:= proc(n) option remember; `if`(n=1, 1, (n-1)!*((-1)^n+add(d*
%p A157159       (-a(d)/d!)^(n/d), d=numtheory[divisors](n) minus {1, n}))
%p A157159        +(-1)^(n+1)*add((k-1)!*Stirling1(n, k), k=1..n))
%p A157159     end:
%p A157159 seq(a(n), n=1..30);  # _Alois P. Heinz_, Aug 14 2012
%t A157159 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_ *)
%Y A157159 Cf. A137852, A006973, A147542.
%K A157159 sign
%O A157159 1,4
%A A157159 _Wolfdieter Lang_ Mar 06 2009
%E A157159 More terms from _Alois P. Heinz_, Aug 14 2012

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE