A036648 Number of centered 5-valent trees with n nodes.

0, 1, 0, 1, 1, 2, 3, 6, 11, 24, 48, 109, 242, 574, 1346, 3258, 7928, 19664, 49158, 124384, 316791, 813231, 2099326, 5451613, 14226697, 37306971, 98247737, 259779698, 689385447, 1835644498, 4902992215, 13133825317, 35276818036

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. A036649, 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;
S5[f_,h_,x_] := f[h,x]^5/120 + f[h,x]^3 f[h,x^2]/12 + f[h,x]^2 f[h,x^3]/6 + f[h,x] f[h,x^2]^2/8 + f[h,x] f[h,x^4]/4 + f[h,x^2] f[h,x^3]/6 + f[h,x^5]/5;
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) + S5[T,h-1,z]z - S5[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 *)

Formula

a(n) = A036650(n) - A036649(n).