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Search: id:A137852
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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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+0
3
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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; listen)
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OFFSET
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1,3
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COMMENT
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Equals signed A006973 (except for initial term), where A006973 lists the
dimensions of representations by Witt vectors.
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FORMULA
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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.
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EXAMPLE
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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!)*...
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PROGRAM
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(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
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Cf. A006973.
Adjacent sequences: A137849 A137850 A137851 this_sequence A137853 A137854 A137855
Sequence in context: A027302 A073981 A006973 this_sequence A097346 A053194 A005582
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KEYWORD
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nice,sign
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Feb 14 2008
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