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A291662 Number of ordered rooted trees with 2n non-root nodes such that the maximal outdegree equals n. 3
1, 1, 8, 53, 326, 1997, 12370, 77513, 490306, 3124541, 20030000, 129024469, 834451788, 5414950283, 35240152706, 229911617041, 1503232609082, 9847379391133, 64617565719052, 424655979547781, 2794563003870310, 18412956934908669, 121455445321173578 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is also the number of permutations p of [2n] such that in 0p the largest up-jump equals n and no down-jump is larger than 1. An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here. a(2) = 8: 1243, 1324, 1342, 2134, 2143, 2314, 2341, 2431.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1208

FORMULA

a(n) = A203717(2n,n).

a(n) ~ (27/4)^n / sqrt(3*Pi*n). - Vaclav Kotesovec, May 02 2018

EXAMPLE

a(2) = 8:

.

.    o      o      o      o      o       o      o       o

.    |      |      |     / \    / \     / \    / \     / \

.    o      o      o    o   o  o   o   o   o  o   o   o   o

.    |     / \    / \   |          |  ( )        ( )  |   |

.    o    o   o  o   o  o          o  o o        o o  o   o

.   / \   |          |  |          |

.  o   o  o          o  o          o

MATHEMATICA

b[n_, t_, k_] := b[n, t, k] = If[n == 0, 1, If[t > 0, Sum[b[j - 1, k, k]* b[n - j, t - 1, k], {j, 1, n}], b[n - 1, k, k]]];

T[n_, k_] := b[n, k - 1, k - 1] - If[k == 1, 0, b[n, k - 2, k - 2]];

a[n_] := T[2n, n];

Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, May 29 2019, after Alois P. Heinz in A203717 *)

PROG

(Python)

from sympy.core.cache import cacheit

@cacheit

def b(u, o, k): return 1 if u + o==0 else sum([b(u - j, o + j - 1, k) for j in range(1, min(1, u) + 1)]) + sum([b(u + j - 1, o - j, k) for j in range(1, min(k, o) + 1)])

def a(n): return b(0, 2*n, n) - (0 if n==0 else b(0, 2*n, n - 1))

print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 30 2017

CROSSREFS

Cf. A203717.

Bisection (even part) of A303259.

Sequence in context: A212754 A099622 A198846 * A110099 A297334 A091870

Adjacent sequences:  A291659 A291660 A291661 * A291663 A291664 A291665

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 28 2017

STATUS

approved

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Last modified June 16 02:14 EDT 2021. Contains 345055 sequences. (Running on oeis4.)