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A000200 Number of bicentered hydrocarbons with n atoms.
(Formerly M2288 N0905)
9
0, 0, 1, 0, 1, 1, 3, 3, 9, 15, 38, 73, 174, 380, 915, 2124, 5134, 12281, 30010, 73401, 181835, 452165, 1133252, 2851710, 7215262, 18326528, 46750268, 119687146, 307528889, 792716193, 2049703887, 5314775856, 13817638615, 36012395538 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

REFERENCES

Busacker and Saaty, Finite Graphs and Networks, 1965, p. 201 (they reproduce Cayley's mistakes).

A. Cayley, "On the mathematical theory of isomers", Phil. Mag. vol. 67 (1874), 444-447.

A. Cayley, "Über die analytischen Figuren, welche in der Mathematik Baeume genannt werden...", Chem. Ber. 8 (1875), 1056-1059.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..60

Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678

H. Bottomley, Illustration of initial terms of A000022, A000200, A000602

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. (Annotated scanned copy)

E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees)., J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.

Index entries for sequences related to trees

N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678

MAPLE

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);

# Maple code continues from A000022: bicentered == unordered pair of ternary trees of the same height:

MATHEMATICA

n = 40; (* algorithm from Rains and Sloane *)

S3[f_, h_, x_] := f[h, x]^3/6 + f[h, x] f[h, x^2]/2 + f[h, x^3]/3;

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];

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 *)

CROSSREFS

A000200 = A000602 - A000022 for n>0.

Cf. A010373.

Sequence in context: A105423 A147471 A062510 * A100744 A285883 A232948

Adjacent sequences:  A000197 A000198 A000199 * A000201 A000202 A000203

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, E. M. Rains (rains(AT)caltech.edu)

STATUS

approved

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Last modified October 19 17:55 EDT 2018. Contains 316376 sequences. (Running on oeis4.)