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A007649 Number of set-like molecular species of degree n.
(Formerly M0824)
2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A327779 A291740 A204520 * 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

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Last modified May 11 13:41 EDT 2021. Contains 343791 sequences. (Running on oeis4.)