A000022 Number of centered hydrocarbons with n atoms. (Formerly M0358 N0135)

0, 1, 0, 1, 1, 2, 2, 6, 9, 20, 37, 86, 181, 422, 943, 2223, 5225, 12613, 30513, 74883, 184484, 458561, 1145406, 2879870, 7274983, 18471060, 47089144, 120528657, 309576725, 797790928, 2062142876, 5345531935, 13893615154, 36201693122

Offset

0,6

References

R. G. Busacker and T. L. Saaty, Finite Graphs and Networks, McGraw-Hill, NY, 1965, p. 201. (They reproduce Cayley's mistakes.)

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.

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

Henry Bottomley, Illustration of initial terms of A000022, A000200, A000602

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 Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)

H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.

H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (8) (1931), 3077-3085.

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.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

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

Index entries for sequences related to trees

Maple

# 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:
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:
for i from 0 to N do c[ i ] := series(q[ i ]-q[ i-1 ]-q2[ i ]+O(z^(N+1)), z, 200); od:
# erase height information: i := 'i': cent := series(sum(c[ i ], i=0..N), z, 200); G000022 := cent; A000022 := n->coeff(G000022, z, n);
# continued in A000200.

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;
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[-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) + 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 *)

Crossrefs

A000022 + A000200 = A000602 for n>0. Cf. A010372.

Sequence in context: A339426 A000021 A367718 * A034805 A192659 A327485

Adjacent sequences: A000019 A000020 A000021 * A000023 A000024 A000025

Keyword

nonn,easy,nice

Author

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

Status

approved