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

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (terms 0..200 from Andrew Howroyd)

M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.

Vaclav Kotesovec, Recurrence (of order 10)

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

a(n) ~ c * d^n / n^(3/2), where d = 4.60104107391278274868305060007404493889954... is the largest positive real root of the equation 16 - 48*d + 4*d^2 + 28*d^3 - 181*d^4 + 976*d^5 - 666*d^6 - 222*d^7 - 69*d^8 - 94*d^9 + 27*d^10 = 0 and c = 0.91053827246762074415410258833655792134... - Vaclav Kotesovec, Feb 08 2026

Mathematica

Join[{1}, Table[Sum[Sum[Sum[(-1)^j * Binomial[k, i] * Binomial[i, j] * Binomial[n-i, k-i-1] * Binomial[n-4*j-1, i-1]/k, {j, 0, Floor[(n-i)/4]}], {i, 1, k-1} ], {k, 1, n+1}], {n, 1, 30}]] (* Vaclav Kotesovec, Feb 08 2026 *)

Prog

(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

Crossrefs

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

Sequence in context: A075316 A192142 A212962 * A047122 A047107 A149040

Adjacent sequences: A246972 A246973 A246974 * A246976 A246977 A246978

Keyword

nonn

Author

N. J. A. Sloane, Sep 14 2014

Extensions

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

Status

approved