A135413 Number of at most 4-way branching ordered (i.e., plane) trees.

1, 2, 6, 20, 70, 246, 875, 3144, 11385, 41470, 151778, 557712, 2056210, 7602700, 28180050, 104677280, 389571983, 1452293766, 5422187130, 20271296100, 75878518695, 284339792110, 1066585128810, 4004566131000, 15048213795600

Offset

1,2

Comments

Obtained by Lagrange inversion of the generating function for at most k-way branching trees.

Solve z = T/(1+T+...T^k) when k = 4. I.e., the n-th term is the coefficient of x^(n-1) in the expansion of (1+x+x^2+x^3+x^4)^n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 250 terms from G. C. Greubel)

Formula

a(n) = [ x^(n-1) ] (1+x+x^2+x^3+x^4)^n.

a(n) = Sum_{i=0..floor((n-1)/5)} (-1)^i * C(n,i) * C(2*n-5*i-2,n-5*i-1). - Vladimir Kruchinin, Mar 28 2019

Maple

A135413 := proc(n) local ogf, i ; ogf := 1 ; for i from 1 to n do ogf := taylor(ogf*(1+x+x^2+x^3+x^4), x=0, n) ; od: coeftayl(ogf, x=0, n-1) ; end: seq(A135413(n), n=1..30) ; # R. J. Mathar, Apr 21 2008

Mathematica

Join[{1}, Table[Coefficient[(1 + x + x^2 + x^3 + x^4)^n, x, (n - 1)], {n, 2, 25}]] (* G. C. Greubel, Oct 13 2016 *)

Prog

(Maxima)
a(n):=sum((-1)^i*binomial(n, i)*binomial(2*n-5*i-2, n-5*i-1), i, 0, (n-1)/5); /* Vladimir Kruchinin, Mar 28 2019 */
(PARI) a(n) = polcoef((1+x+x^2+x^3+x^4)^n, n-1, x); \\ Michel Marcus, Mar 28 2019

Crossrefs

For k=2 this is A005717, for k=3 this is A005726.

Sequence in context: A045631 A275046 A229472 * A193653 A147748 A150125

Adjacent sequences: A135410 A135411 A135412 * A135414 A135415 A135416

Keyword

nonn,easy

Author

Andrey Bovykin (indiscernibles(AT)googlemail.com), Mar 01 2008

Extensions

More terms from R. J. Mathar, Apr 21 2008

Status

approved