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A006973 Dimensions of representations by Witt vectors.
(Formerly M1921)
25
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 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).

LINKS

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

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

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

Johann Cigler, Some remarks on the power product expansion of the q-exponential series, arXiv:2006.06242 [math.CO], 2020.

Gottfried Helms, A dream of a (number-) sequence, 2007-2009.

C. Reutenauer, 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. [Annotated scanned copy]

Christophe Reutenauer, Sur des fonctions symétriques reliées 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 symétriques reliées 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_{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

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: A213720 A353822 A073981 * A137852 A347106 A097346

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 July 3 16:27 EDT 2022. Contains 355055 sequences. (Running on oeis4.)