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).
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
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, E. M. Rains (rains(AT)caltech.edu)
STATUS
approved