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A137852 G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(x). 7
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

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-Fran├žois Alcover, Nov 28 2011 *)

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: A213720 A073981 A006973 * A097346 A261431 A226388

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 October 23 21:59 EDT 2017. Contains 293814 sequences.