A007649 Number of set-like molecular species of degree n. (Formerly M0824)

1, 1, 2, 3, 7, 9, 20, 26, 54, 74, 137, 184, 356, 473, 841, 1154, 2034, 2742, 4740, 6405, 10874, 14794, 24515, 33246, 54955, 74380, 120501, 163828, 263144, 356621, 567330, 768854, 1212354, 1644335, 2567636, 3478873, 5403223, 7314662, 11265825, 15258443, 23363143, 31608055, 48113280, 65063640, 98501538, 133168305, 200503864

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

Cf. A000638, A007650.

Sequence in context: A291740 A204520 A358392 * A218909 A225202 A046152

Adjacent sequences: A007646 A007647 A007648 * A007650 A007651 A007652

Keyword

nonn

Author

Simon Plouffe

Extensions

Added more terms, Joerg Arndt, Jul 30 2012

Status

approved