A036649 Number of bicentered 5-valent trees with n nodes.

0, 0, 1, 0, 1, 1, 3, 4, 10, 18, 46, 95, 231, 524, 1287, 3095, 7713, 19125, 48258, 122026, 311935, 801061, 2072629, 5387753, 14081981, 36959506, 97419796, 257724555, 684254908, 1822560590, 4869517848, 13047469920, 35053803135

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• 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

Crossrefs

Cf. A036648, A036650.

Programs

Mathematica

n = 30; (* algorithm from Rains and Sloane *)
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 + S4[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 *)

Formula

a(n) = A036650(n) - A036648(n).