A246975 Number of 4-ary plane multitrees with n edges.

1, 1, 3, 10, 36, 121, 447, 1699, 6589, 25914, 103633, 419421, 1714463, 7068285, 29361629, 122764876, 516245009, 2181957489, 9264275600, 39495666700, 169000837410, 725574719515, 3124648750706, 13493792787415, 58422790497226, 253547380435914, 1102776319943605

Offset: 0

Author

N. J. A. Sloane, Sep 14 2014

Keywords

• nonn

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. A128720 (2-ary case), A246974 (3-ary case).

Programs

PARI

a(n)={my(m=4); 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)/4)} (-1)^j*binomial(k, i)*binomial(i, j)*binomial(n-i, k-i-1)*binomial(n-4*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