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A000022 Number of centered hydrocarbons with n atoms.

%I A000022 M0358 N0135 #48 Dec 19 2020 03:13:30
%S A000022 0,1,0,1,1,2,2,6,9,20,37,86,181,422,943,2223,5225,12613,30513,74883,
%T A000022 184484,458561,1145406,2879870,7274983,18471060,47089144,120528657,
%U A000022 309576725,797790928,2062142876,5345531935,13893615154,36201693122
%N A000022 Number of centered hydrocarbons with n atoms.
%D A000022 R. G. Busacker and T. L. Saaty, Finite Graphs and Networks, McGraw-Hill, NY, 1965, p. 201. (They reproduce Cayley's mistakes.)
%D A000022 A. Cayley, "Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen", Chem. Ber. 8 (1875), 1056-1059.
%D A000022 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000022 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000022 N. J. A. Sloane, <a href="/A000022/b000022.txt">Table of n, a(n) for n = 0..60</a>
%H A000022 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 A000022 Henry Bottomley, <a href="/A000602/a000602.gif">Illustration of initial terms of A000022, A000200, A000602</a>
%H A000022 A. Cayley, <a href="http://www.archive.org/stream/collectedmathema09cayl#page/202">On the mathematical theory of isomers</a>, Phil. Mag. vol. 67 (1874), 444-447.
%H A000022 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 A000022 H. R. Henze and C. M. Blair, <a href="http://dx.doi.org/10.1021/ja01359a027">The number of structurally isomeric alcohols of the methanol series</a>, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.
%H A000022 H. R. Henze and C. M. Blair, <a href="http://dx.doi.org/10.1021/ja01359a034">The number of isomeric hydrocarbons of the methane series</a>, J. Amer. Chem. Soc., 53 (8) (1931), 3077-3085.
%H A000022 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 A000022 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000022 N. J. A. Sloane, <a href="/A000602/a000602.txt">Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678</a>
%H A000022 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%p A000022 # We continue from the Maple code in A000678: Unordered 4-tuples of ternary trees with one of height i and others of height at most i-1:
%p A000022 N := 45: i := 1: while i<(N+1) do Tb := t[ i ]-t[ i-1 ]: Ts := t[ i ]-1: Q2 := series(Tb*Ts+O(z^(N+1)),z,200): q2[ i ] := Q2: i := i+1; od: q2[ 0 ] := 0: q[ -1 ] := 0:
%p A000022 for i from 0 to N do c[ i ] := series(q[ i ]-q[ i-1 ]-q2[ i ]+O(z^(N+1)),z,200); od:
%p A000022 # erase height information: i := 'i': cent := series(sum(c[ i ],i=0..N),z,200); G000022 := cent; A000022 := n->coeff(G000022,z,n);
%p A000022 # continued in A000200.
%t A000022 n = 40; (* algorithm from Rains and Sloane *)
%t A000022 S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
%t A000022 S4[f_,h_,x_] := f[h,x]^4/24 + f[h,x]^2 f[h,x^2]/4 + f[h,x] f[h,x^3]/3 + f[h,x^2]^2/8 + f[h,x^4]/4;
%t A000022 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 A000022 Sum[Take[CoefficientList[z^(n+1) + S4[T,h-1,z]z - S4[T,h-2,z]z - (T[h-1,z] - T[h-2,z]) (T[h-1,z]-1),z], n+1], {h,1,n/2}] + PadRight[{0,1}, n+1] (* _Robert A. Russell_, Sep 15 2018 *)
%Y A000022 A000022 + A000200 = A000602 for n>0. Cf. A010372.
%K A000022 nonn,easy,nice
%O A000022 0,6
%A A000022 _N. J. A. Sloane_, E. M. Rains (rains(AT)caltech.edu)