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A006973 Dimensions of representations by Witt vectors.
(Formerly M1921)
8
0, 1, 2, 9, 24, 130, 720, 8505, 35840, 412776, 3628800, 42030450, 479001600, 7019298000, 82614884352, 1886805545625, 20922789888000, 374426276224000, 6402373705728000, 134987215801622184, 2379913632645120000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Starting (1, 2, 9, 24,...) = row sums of triangle A156792. - Gary W. Adamson, Feb 15 2009

REFERENCES

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, Mar 26 1992.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

J. Borwein, Letter to C. Reutenauer, n.d.

Jonathan Borwein, Shi Tuo Lou, Asymptotics of a sequence of Witt vectors, J. Approx. Theory 69 (1992), no. 3, 326-337. Math. Rev. 93f:05007.

C. Reutenauer, 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, Mar 26 1992. [Annotated scanned copy]

Christophe Reutenauer, 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.

C. Reutenauer, 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. (Annotated scanned copy)

FORMULA

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(sum(M1(fp)*product(a(k[j]),j=1..m),fp from FP(n,m)),m=2..maxm(n)), 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(1 + a(j)*(x^j)/j!,j=1..infinity). 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

EXAMPLE

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 + ...

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A137852, A156792.

Sequence in context: A027302 A213720 A073981 * A137852 A097346 A261431

Adjacent sequences:  A006970 A006971 A006972 * A006974 A006975 A006976

KEYWORD

nonn,easy,nice

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Michael Somos, Oct 07 2001

Further terms from Paul D. Hanna, Feb 14 2008

STATUS

approved

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Last modified January 17 05:26 EST 2019. Contains 319207 sequences. (Running on oeis4.)