A301740 The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.

3, 9, 24, 50, 96, 164, 267, 408, 603, 856, 1186, 1598, 2115, 2742, 3505, 4411, 5489, 6746, 8215, 9904, 11849, 14059, 16573, 19401, 22586, 26138, 30103, 34493, 39357, 44707, 50596, 57037, 64086, 71757, 80109, 89157, 98964, 109545, 120966, 133244, 146448, 160595, 175758, 191955

Offset: 5

Author

R. J. Mathar, Mar 26 2018

Keywords

• nonn
• easy

Comments

Computed by the sum over the A000055(5)=3 shapes of the trees: the linear graph of the n-Pentane, the branched 2-Methyl-Butane, and the star graph of (1,1)-Bimethyl-Propane.

Examples

			a(5)=3 because there is a linear tree with all labels equal 1, the branched tree with all labels equal to 1, and the star tree with all labels equal 1.
		

Links

• R. J. Mathar, Labeled Trees with Fixed Node Label sum, sequence v_5.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).

Crossrefs

Cf. A002620 (labeled trees with 3 nodes), A301739 (labeled trees with 4 nodes).

Programs

Maple

-x^5*(3+3*x+6*x^2+5*x^3+5*x^4+2*x^5+x^6)/(1+x^2)/(1+x+x^2)/(1+x)^2/(x-1)^5 ;
taylor(%,x=0,80) ;
gfun[seriestolist](%) ;

Mathematica

LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,-2,1},{3,9,24,50,96,164,267,408,603,856,1186},50] (* Harvey P. Dale, Jun 16 2025 *)

Formula

a(n) = A005994(n-5)+A001752(n-5)+A002621(n-5).