A246974 Number of 3-ary plane multitrees with n edges.

1, 1, 3, 10, 28, 93, 317, 1090, 3876, 13995, 51182, 189606, 709218, 2675230, 10166639, 38883721, 149559230, 578153160, 2245017535, 8752828951, 34250020397, 134465596581, 529509173245, 2090920335200, 8277633788511, 32846871639751, 130624556118075, 520512049658200

Offset: 0

Author

N. J. A. Sloane, Sep 14 2014

Keywords

• nonn

Comments

A k-ary plane multitree is a plane tree with edges having multiplicity and the outdegree of any node does not exceed k. The number of plane multitrees with n edges (without restriction on outdegree) is given by A002212(n). - Andrew Howroyd, Feb 24 2020

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.

Crossrefs

Cf. A002212, A128720 (2-ary case), A246975 (4-ary case).

Programs

PARI

a(n)={my(m=3); if(n<1, n==0, sum(k=1, n+1, sum(i=1, k-1, sum(j=0, (n-i)\m, (-1)^j*binomial(k, i)*binomial(i, j)*binomial(n-i, k-i-1)*binomial(n-m*j-1, i-1)))/k))} \\ Andrew Howroyd, Feb 24 2020

Formula

a(n) = Sum_{k=1..n+1} Sum_{i=1..k-1} Sum_{j=0..floor((n-i)/3)} (-1)^j*binomial(k, i)*binomial(i, j)*binomial(n-i, k-i-1)*binomial(n-3*j-1, i-1)/k for n > 0. - Andrew Howroyd, Feb 24 2020

Extensions

Terms a(11) and beyond from Andrew Howroyd, Feb 24 2020