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%I M2288 N0905 #40 Dec 19 2020 03:13:09
%S 0,0,1,0,1,1,3,3,9,15,38,73,174,380,915,2124,5134,12281,30010,73401,
%T 181835,452165,1133252,2851710,7215262,18326528,46750268,119687146,
%U 307528889,792716193,2049703887,5314775856,13817638615,36012395538
%N Number of bicentered hydrocarbons with n atoms.
%D Busacker and Saaty, Finite Graphs and Networks, 1965, p. 201 (they reproduce Cayley's mistakes).
%D A. Cayley, "On the mathematical theory of isomers", Phil. Mag. vol. 67 (1874), 444-447.
%D A. Cayley, "Über die analytischen Figuren, welche in der Mathematik Baeume genannt werden...", Chem. Ber. 8 (1875), 1056-1059.
%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="/A000200/b000200.txt">Table of n, a(n) for n = 0..60</a>
%H Jean-François Alcover, <a href="/A000602/a000602_1.txt">Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678</a>
%H Henry Bottomley, <a href="/A000602/a000602.gif">Illustration of initial terms of A000022, A000200, A000602</a>
%H A. Cayley, <a href="/A000022/a000022.pdf">Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen</a>, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
%H E. M. Rains and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/cayley.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees)</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H N. J. A. Sloane, <a href="/A000602/a000602.txt">Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%p N := 45: for i from 1 to N do tt := t[ i ]-t[ i-1 ]; b[ i ] := series((tt^2+subs(z=z^2,tt))/2+O(z^(N+1)),z,200): od: i := 'i': bicent := series(sum(b[ i ],i=1..N),z,200); G000200 := bicent; A000200 := n->coeff(G000200,z,n);
%p # Maple code continues from A000022: bicentered == unordered pair of ternary trees of the same height:
%t n = 40; (* algorithm from Rains and Sloane *)
%t S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
%t T[-1,z_] := 1; T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S3[T,h-1,z]z, z], n+1];
%t Sum[Take[CoefficientList[z^(n+1) + (T[h,z] - T[h-1,z])^2/2 + (T[h,z^2] - T[h-1,z^2])/2, z],n+1], {h,0,n/2}] (* _Robert A. Russell_, Sep 15 2018 *)
%Y A000200 = A000602 - A000022 for n>0.
%Y Cf. A010373.
%K nonn,nice
%O 0,7
%A _N. J. A. Sloane_, E. M. Rains (rains(AT)caltech.edu)
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