1,3

Starting (1, 2, 9, 24,...) = row sums of triangle A156792. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 15 2009]

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

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

Reutenauer, Christophe; 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.

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, 26 March 1992.

G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = exp(-x)/(1-x). - Paul D. Hanna (pauldhanna(AT)juno.com), 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. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 24 2009]

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. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 24 2009]

(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 (pauldhanna(AT)juno.com), Feb 14 2008

Cf. A137852.

A156792 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 15 2009]

Sequence in context: A143561 A027302 A073981 this_sequence A137852 A097346 A053194

Adjacent sequences: A006970 A006971 A006972 this_sequence A006974 A006975 A006976

nonn,easy,nice

Simon Plouffe (simon.plouffe(AT)gmail.com)

More terms from Michael Somos, Oct 07, 2001

More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 14 2008