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A006973
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Dimensions of representations by Witt vectors.
(Formerly M1921)
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25
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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
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OFFSET
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1,3
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COMMENTS
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REFERENCES
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Reutenauer, Christophe; Sur des fonctions symétriques liées aux vecteurs de Witt et à l'algèbre de Lie libre, Report 177, Dept. Mathématiques et d'Informatique, Univ. Québec à Montréal, Mar 26 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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, Feb 14 2008
A 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 M_1 array for any partition in A036038): a(n) = (-1)^n - Sum_{m=2..maxm(n)} ( Sum_{fp from FP(n,m)} (M1(fp)*Product_{j=1..m} ( a(k[j]) ) ), with maxm(n) = A003056(n) = floor((sqrt(1+8*n) -1)/2) and the distinct parts k[j], j=1..m, of the partition of n, n>=2, with input a(1)=-1 (but only for this recurrence). Note that a(1)=0. Proof by comparing coefficients of (x^n)/n! in exp(-x) = (1-x)*Product_{j>=1} ( 1 + a(j)*(x^j)/j! ). See array A008289(n,m) for the cardinality of the set FP(n,m). Another recurrence has been given in the first PARI program line below. - Wolfdieter Lang, Feb 24 2009
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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!)*...
Recurrence: a(7) = -1 - (7*a(1)*a(6) + 21*a(2)*a(5) + 35 a(3)*a(4) + 105*a(1)*a(2)*a(4)) = -1 -(-910 + 504 + 630 - 945) = 720 = 6!. For the recurrence one has to use a(1)=-1. - Wolfdieter Lang, Feb 24 2009
G.f. = x^2 + 2*x^3 + 9*x^4 + 24*x^5 + 130*x^6 + 720*x^7 + 8505*x^8 + ...
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MATHEMATICA
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a[n_] := a[n] = If[n < 4, Max[n-1, 0], (n-1)!*(1 + Sum[ k*(-a[k]/k!)^(n/k), {k, Most[Divisors[n]]}])]; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Jul 19 2012, after 1st PARI program *)
a[ n_]:= If[n<2, 0, a[n] = n! SeriesCoefficient[ Exp[-x]/((1-x) Product[ 1 + a[k] x^k/k!, {k, 2, n-1}]), {x, 0, n}]]; (* Michael Somos, Feb 23 2015 *)
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PROG
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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, Feb 14 2008
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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