OFFSET
0,3
COMMENTS
A set-like molecular species is formed from the species of sets, E, through species addition, product and substitution.
REFERENCES
G. Labelle and P. Leroux, Identities and enumeration: weighting connected components, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Joerg Arndt, Table of n, a(n) for n = 0..100
G. Labelle and P. Leroux, Identities and enumeration: weighting connected components, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896. (Annotated scanned copy)
G. Labelle and P. Leroux, An extension of the exponential formula in enumerative combinatorics, The Electronic Journal of Combinatorics, Volume 3, Issue 2 (1996) (The Foata Festschrift volume), Research Paper #R12.
N. J. A. Sloane, Transforms
FORMULA
Euler Transform of A007650. Define c(n): c(0)=0. c(k)=a(k), k>0. A007650=MOEBIUSi(c)-c. - Christian G. Bower, Feb 23 2006
MATHEMATICA
NN = 66; va = Array[0&, NN]; va[[1]] = 0; va[[2]] = 1; vm = Array[0&, NN]; vm[[1]] = 1; vm[[2]] = 1; For[n = 2, n <= NN - 1, n++, va[[n+1]] = DivisorSum[n , vm[[#+1]]&]; vm[[n+1]] = 1/n*Sum[DivisorSum[k, #*va[[#+1]] &]*vm[[n-k+1]], {k, 1, n}]]; vm (* Jean-François Alcover, Dec 01 2015, adapted from Joerg Arndt's PARI script *)
PROG
(PARI) /* From the Labelle/Leroux reference */
N=66; /* that many terms */
va=vector(N); va[0+1] = 0; va[1+1] = 1;
vm=vector(N); vm[0+1] = 1; vm[1+1] = 1;
{ for(n=2, N-1,
va[n+1] = sumdiv(n, k, vm[k+1]);
vm[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*va[d+1]) * vm[n-k+1] );
); }
v007649=vm
/* v007650=va */
/* Joerg Arndt, Jul 30 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Added more terms, Joerg Arndt, Jul 30 2012
STATUS
approved