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 A137852 G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(x). 24
 1, 1, -2, 9, -24, 130, -720, 8505, -35840, 412776, -3628800, 42030450, -479001600, 7019298000, -82614884352, 1886805545625, -20922789888000, 374426276224000, -6402373705728000, 134987215801622184, -2379913632645120000, 55685679780013920000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Equals signed A006973 (except for initial term), where A006973 lists the dimensions of representations by Witt vectors. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..170 Gottfried Helms, A dream of a (number-) sequence, 2007-2009. FORMULA a(n) = (n-1)!*[(-1)^n + Sum_{d divides n, 11 with a(1)=1. Another recurrence. 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 fp(n,m) from FP(n,m): a(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(n,m). Inputs a(1)=1, a(2)=1. See also array A008289(n,m) for the cardinality of the set FP(n,m). - Wolfdieter Lang, Feb 20 2009 EXAMPLE exp(x) = (1+x)*(1+x^2/2!)*(1-2*x^3/3!)*(1+9*x^4/4!)*(1-24*x^5/5!)* (1+130*x^6/6!)*(1-720*x^7/7!)*(1+8505*x^8/8!)*(1-35840*x^9/9!)*(1+412776*x^10/10!)*(1-3628800*x^11/11!)*...*(1+a(n)*x^n/n!)*... Another recurrence: n=6; m=1,2,3=maxm(6)=A003056(6); fp(6,2) from {(1,5),(2,4)}, fp(6,3)=(1,2,3); a(6)= 1 - ( 6*a(1)*a(5) + 15*a(2)*a(4) + 60*a(1)*a(2)*a(3)). Check: 1 - (6*1*(-24) + 15*1*9 +60*1*1*(-2)) = 130 = a(6). - Wolfdieter Lang, Feb 20 2009 MAPLE with(numtheory): 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}))) end: seq(a(n), n=1..30); # Alois P. Heinz, Aug 14 2012 MATHEMATICA max = 22; f[x_] := Product[1 + a[n] x^n/n!, {n, 1, max}]; coes = CoefficientList[ Series[f[x] - Exp[x], {x, 0, max}], x]; sol = Solve[ Thread[coes == 0]][[1]]; Table[a[n] /. sol, {n, 1, max}] (* Jean-François Alcover, Nov 28 2011 *) a[1] = 1; a[n_] := a[n] = (n-1)!*((-1)^n + Sum[d*(-a[d]/d!)^(n/d), {d, Divisors[n] ~Complement~ {1, n}}]); Array[a, 30] (* Jean-François Alcover, Jan 11 2018 *) PROG (PARI) {a(n)=if(n<1, 0, if(n==1, 1, (n-1)!*((-1)^n + sumdiv(n, d, if(d1, d*(-a(d)/d!)^(n/d))))))} for(n=1, 30, print1(a(n), ", ")) (PARI) /* As coefficients in product g.f.: */ {a(n)=if(n<1, 0, n!*polcoeff(exp(x +x*O(x^n))/prod(k=0, n-1, 1+a(k)*x^k/k! +x*O(x^n)), n))} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A006973. Sequence in context: A353822 A073981 A006973 * A347106 A097346 A343576 Adjacent sequences: A137849 A137850 A137851 * A137853 A137854 A137855 KEYWORD nice,sign AUTHOR Paul D. Hanna, Feb 14 2008 STATUS approved

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