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%I M1921 #55 Jun 12 2021 23:20:21
%S 0,1,2,9,24,130,720,8505,35840,412776,3628800,42030450,479001600,
%T 7019298000,82614884352,1886805545625,20922789888000,374426276224000,
%U 6402373705728000,134987215801622184,2379913632645120000
%N Dimensions of representations by Witt vectors.
%C Starting (1, 2, 9, 24, ...) = row sums of triangle A156792. - _Gary W. Adamson_, Feb 15 2009
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006973/b006973.txt">Table of n, a(n) for n = 1..200</a>
%H J. Borwein, <a href="/A006973/a006973_1.pdf">Letter to C. Reutenauer, n.d.</a>
%H Jonathan Borwein and Shi Tuo Lou, <a href="http://dx.doi.org/10.1016/0021-9045(92)90006-A">Asymptotics of a sequence of Witt vectors</a>, J. Approx. Theory 69 (1992), no. 3, 326-337. Math. Rev. 93f:05007.
%H Johann Cigler, <a href="https://arxiv.org/abs/2006.06242">Some remarks on the power product expansion of the q-exponential series</a>, arXiv:2006.06242 [math.CO], 2020.
%H Gottfried Helms, <a href="http://go.helms-net.de/math/musings/dreamofasequence.pdf">A dream of a (number-) sequence</a>, 2007-2009.
%H C. Reutenauer, <a href="/A006973/a006973_2.pdf">Sur des fonctions symétriques liées aux vecteurs de Witt et à l'algèbre de Lie libre</a>, Report 177, Dept. Mathématiques et d'Informatique, Univ. Québec à Montréal, Mar 26 1992. [Annotated scanned copy]
%H Christophe Reutenauer, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k54707877/f493.image">Sur des fonctions symétriques reliées aux vecteurs de Witt</a>, [ On symmetric functions related to Witt vectors ] C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), no. 7, 487-490.
%H C. Reutenauer, <a href="/A006973/a006973.pdf">Sur des fonctions symétriques reliées aux vecteurs de Witt, [ On symmetric functions related to Witt vectors ] </a>, C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), no. 7, 487-490. (Annotated scanned copy)
%F G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(-x)/(1-x). - _Paul D. Hanna_, Feb 14 2008
%F 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
%e G.f.: exp(-x)/(1-x) = (1 + 0*x)*(1 + 1*x^2/2!)*(1 + 2*x^3/3!)*(1 + 9*x^4/4!)*
%e (1 + 24*x^5/5!)*(1 + 130*x^6/6!)*...*(1 + a(n)*x^n/n!)*...
%e 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
%e G.f. = x^2 + 2*x^3 + 9*x^4 + 24*x^5 + 130*x^6 + 720*x^7 + 8505*x^8 + ...
%t 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 *)
%t 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 *)
%o (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)))))
%o (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
%Y Cf. A137852, A156792.
%K nonn,easy,nice
%O 1,3
%A _Simon Plouffe_
%E More terms from _Michael Somos_, Oct 07 2001
%E Further terms from _Paul D. Hanna_, Feb 14 2008
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