A006973 Dimensions of representations by Witt vectors.
0, 1, 2, 9, 24, 130, 720, 8505, 35840, 412776, 3628800, 42030450, 479001600, 7019298000, 82614884352, 1886805545625, 20922789888000, 374426276224000, 6402373705728000, 134987215801622184, 2379913632645120000
Offset: 1
Examples
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 + ...
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)
Programs
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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 *) -
PARI
a(n)=if(n<4,max(n-1,0),(n-1)!*(1+sumdiv(n,k, if(k
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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
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
Extensions
More terms from Michael Somos, Oct 07 2001
Further terms from Paul D. Hanna, Feb 14 2008
Comments