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A137852
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G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(x).
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6
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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; internal format)
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OFFSET
| 1,3
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COMMENTS
| Equals signed A006973 (except for initial term), where A006973 lists the
dimensions of representations by Witt vectors.
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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). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 20 2009]
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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). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 20 2009]
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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}] (* From Jean-François Alcover, Nov 28 2011 *)
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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))))))} (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))}
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CROSSREFS
| Cf. A006973.
Sequence in context: A027302 A073981 A006973 * A097346 A053194 A005582
Adjacent sequences: A137849 A137850 A137851 * A137853 A137854 A137855
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KEYWORD
| nice,sign
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Feb 14 2008
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