A007649 Number of set-like molecular species of degree n.
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
Keywords
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.
Programs
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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 *) -
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 */
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
Extensions
Added more terms, Joerg Arndt, Jul 30 2012
Comments