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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, 1<d<n} d*( -a(d)/d! )^(n/d) ] for n>1 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(d<n&d>1, 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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Last modified September 27 12:49 EDT 2022. Contains 357057 sequences. (Running on oeis4.)