cp's OEIS Frontend

A007649 Number of set-like molecular species of degree n.

%I A007649 M0824 #33 Dec 31 2024 06:32:40
%S A007649 1,1,2,3,7,9,20,26,54,74,137,184,356,473,841,1154,2034,2742,4740,6405,
%T A007649 10874,14794,24515,33246,54955,74380,120501,163828,263144,356621,
%U A007649 567330,768854,1212354,1644335,2567636,3478873,5403223,7314662,11265825,15258443,23363143,31608055,48113280,65063640,98501538,133168305,200503864
%N A007649 Number of set-like molecular species of degree n.
%C A007649 A set-like molecular species is formed from the species of sets, E, through species addition, product and substitution.
%D A007649 G. Labelle and P. Leroux, Identities and enumeration: weighting connected components, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896.
%D A007649 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007649 Joerg Arndt, <a href="/A007649/b007649.txt">Table of n, a(n) for n = 0..100</a>
%H A007649 G. Labelle and P. Leroux, <a href="/A007649/a007649.pdf">Identities and enumeration: weighting connected components</a>, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896. (Annotated scanned copy)
%H A007649 G. Labelle and P. Leroux, <a href="https://doi.org/10.37236/1270">An extension of the exponential formula in enumerative combinatorics</a>, The Electronic Journal of Combinatorics, Volume 3, Issue 2 (1996) (The Foata Festschrift volume), Research Paper #R12.
%H A007649 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A007649 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
%t A007649 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 *)
%o A007649 (PARI) /* From the Labelle/Leroux reference */
%o A007649 N=66; /* that many terms */
%o A007649 va=vector(N);  va[0+1] = 0; va[1+1] = 1;
%o A007649 vm=vector(N);  vm[0+1] = 1; vm[1+1] = 1;
%o A007649 { for(n=2, N-1,
%o A007649     va[n+1] = sumdiv(n, k, vm[k+1]);
%o A007649     vm[n+1] = 1/n * sum(k=1, n,  sumdiv(k, d, d*va[d+1]) * vm[n-k+1] );
%o A007649 ); }
%o A007649 v007649=vm
%o A007649 /* v007650=va */
%o A007649 /* _Joerg Arndt_, Jul 30 2012 */
%Y A007649 Cf. A000638, A007650.
%K A007649 nonn
%O A007649 0,3
%A A007649 _Simon Plouffe_
%E A007649 Added more terms, _Joerg Arndt_, Jul 30 2012