%I M3466 N1410 #27 Dec 19 2021 09:41:06
%S 0,1,4,13,42,131,402,1218,3657,10899,32298,95257,279844,819390,
%T 2392392,6967956,20250974,58744089,170118980,491913999,1420493862,
%U 4096940530,11803172152,33970257473,97678027311,280624328431,805587723862
%N Number of paraffins C_n H_{2n-1} XYZ with n carbon atoms.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A000640/b000640.txt">Table of n, a(n) for n = 0..100</a>
%H Frederic Chyzak, <a href="http://algo.inria.fr/libraries/autocomb/Polya-html/Polya.html">Enumerating alcohols and other classes of chemical molecules</a>
%H G. Polya, <a href="http://dx.doi.org/10.1524/zkri.1936.93.1.415">Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen</a>, Zeit. f. Kristall., 93 (1936), 415-443.
%H G. Polya, <a href="/A000598/a000598_3.pdf">Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen</a>, Zeit. f. Kristall., 93 (1936), 415-443. (Annotated scanned copy)
%F G.f.: A(x) = x*A000598(x)/(1-A000642(x))^3.
%p # The following Maple commands are taken from the Chyzak web site:
%p with(combstruct);
%p gramm_Alkyl:=Alkyl=Prod(Carbon,Set(Alkyl,card<=3)),Carbon=Atom:
%p specs_Alkyl:=[Alkyl,{gramm_Alkyl},unlabeled]:
%p gramm_S1_Alkyl:=S1_Alkyl[X]=Union(Prod(Carbon,S1_Alkyl[X],Set(Alkyl,card<=2)),Prod(Prod(Carbon,X),Set(Alkyl,card<=2))),X=Epsilon:
%p specs_S1_Alkyl:=[S1_Alkyl[X],{gramm_S1_Alkyl,gramm_Alkyl},unlabeled]:
%p gramm_S2_Alkyl:=S2_Alkyl[X,Y]=Union(Prod(Carbon,S2_Alkyl[X,Y], Set(Alkyl,card<=2)),Prod(Carbon,Union(S1_Alkyl[X],X),Union(S1_Alkyl[Y],Y),Set(Alkyl,card<=1))):
%p specs_S2_Alkyl:=[S2_Alkyl[X,Y],{gramm_S2_Alkyl,gramm_S1_Alkyl,op(subs(X=Y,[gramm_S1_Alkyl])),gramm_Alkyl},unlabeled]:
%p [seq(count(specs_S2_Alkyl,size=i),i=0..50)];
%t terms = 27; (* B,B2 = g.f. for A000598,A000642 resp. *) B[_] = 0; Do[B[x_] = 1 + (1/6)*x*(B[x]^3 + 3*B[x]*B[x^2] + 2*B[x^3]) + O[x]^terms // Normal, terms];
%t B2[x_] = (1/2)*x*(B[x^2] + B[x]^2) + O[x]^terms;
%t A[x_] = x*B[x]/(1 - B2[x])^3 + O[x]^terms;
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 10 2018 *)
%Y Cf. A000598, A000642.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_