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

5,1

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.

Links

Table of n, a(n) for n=5..48.

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).

Formula

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

Example

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.

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](%) ;

Crossrefs

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

Sequence in context: A120012 A352640 A029530 * A227018 A244504 A085739

Adjacent sequences: A301737 A301738 A301739 * A301741 A301742 A301743

Keyword

nonn,easy

Author

R. J. Mathar, Mar 26 2018

Status

approved