A000022 Number of centered hydrocarbons with n atoms.

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

Author

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

Keywords

• nonn
• easy
• nice

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

Crossrefs

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

Programs

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 *)