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Search: id:A006973
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| A006973 |
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Dimensions of representations by Witt vectors.
(Formerly M1921)
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+0
4
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| 0, 1, 2, 9, 24, 130, 720, 8505, 35840, 412776, 3628800, 42030450, 479001600, 7019298000, 82614884352, 1886805545625, 20922789888000, 374426276224000, 6402373705728000, 134987215801622184, 2379913632645120000
(list; graph; listen)
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OFFSET
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1,3
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REFERENCES
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Borwein, Jonathan; Lou, Shi Tuo, Asymptotics of a sequence of Witt vectors. J. Approx. Theory 69 (1992), no. 3, 326-337. Math. Rev. 93f:05007
Reutenauer, Christophe; Sur des fonctions symetriques reliees aux vecteurs de Witt. [ On symmetric functions related to Witt vectors ] C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), no. 7, 487-490.
Reutenauer, Christophe; Sur des fonctions symetriques liees aux vecteurs de Witt et a l'algebre de Lie libre, Report 177, Dept. Mathematiques et d'Informatique, Univ. Quebec a Montreal, 26 March 1992.
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FORMULA
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G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(-x)/(1-x). - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 14 2008
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EXAMPLE
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G.f.: exp(-x)/(1-x) = (1+0*x)*(1+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 + a(n)*x^n/n!)*...
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PROGRAM
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(PARI) a(n)=if(n<4, max(n-1, 0), (n-1)!*(1+sumdiv(n, k, if(k<n, k*(-a(k)/k!)^(n/k)))))
(PARI) /* As coefficients in product g.f.: */ {a(n)=if(n<2, 0, n!*polcoeff((exp(-x+x*O(x^n))/(1-x))/prod(k=0, n-1, 1+a(k)*x^k/k! +x*O(x^n)), n))} - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 14 2008
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CROSSREFS
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Cf. A137852.
Sequence in context: A122006 A027302 A073981 this_sequence A137852 A097346 A053194
Adjacent sequences: A006970 A006971 A006972 this_sequence A006974 A006975 A006976
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Simon Plouffe (plouffe(AT)math.uqam.ca)
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EXTENSIONS
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More terms from Michael Somos, Oct 07, 2001
More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 14 2008
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