A057547 A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

2, 12, 52, 56, 212, 216, 228, 232, 240, 852, 856, 868, 872, 880, 916, 920, 932, 936, 944, 964, 968, 976, 992, 3412, 3416, 3428, 3432, 3440, 3476, 3480, 3492, 3496, 3504, 3524, 3528, 3536, 3552, 3668, 3672, 3684, 3688, 3696, 3732, 3736, 3748, 3752, 3760

Offset

0,1

Comments

This one-to-one correspondence between all rooted plane trees and one node larger, root degree = 1 trees illustrates the fact that INVERT(A000108) = LEFT(A000108). (Catalan numbers shift left under Cameron's A transformation.)

From Ruud H.G. van Tol, May 13 2024: (Start)

Sequence on a lattice:

Tree Paths Decimal Count

|_ 10 2 1

|_._ 1100 12 1

|_|_._ 110100 -111000 52,56 2

|_|_|_._ 11010100 -11110000 212-240 5

|_|_|_|_._ 1101010100-1111100000 852-992 14

... (End)

Links

Ruud H.G. van Tol, Table of n, a(n) for n = 0..2055

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.

P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

Index entries for encodings of plane rooted trees

Formula

a(n) = A014486(A057548(n)) and also from n > 0 onward = A079946(A014486(n)).

a(n) = alltrees2singletrunked(A014486[n]) (see Maple code below and in A057501).

Maple

alltrees2singletrunked := n -> pars2binexp([binexp2pars(n)]); # Just surround with extra parentheses.

Prog

(PARI) a_rows(N) = my(a=Vec([[2]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, for(i=1, valuation(t, 2), listput(~c, (t<<2)+(2<<i)))); a[r+1]=Vec(c)); a; \\ Ruud H.G. van Tol, May 25 2024

Crossrefs

Double-trunked trees: A057517. Cf. also A057548, A057549.

Sequence in context: A323851 A054667 A009537 * A216648 A043007 A300572

Adjacent sequences: A057544 A057545 A057546 * A057548 A057549 A057550

Keyword

nonn,base

Author

Antti Karttunen Sep 07 2000

Status

approved