A073202 -id:A073202 - cp's OEIS frontend

A079227 Number of Catalan objects fixed by six-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

1, 1, 2, 5, 14, 38, 111, 332, 1029, 3232, 10374, 33679, 110722, 367252, 1228558, 4138120, 14025473, 47792389, 163643066, 562722427, 1942548520, 6729230281, 23385132060, 81503084084, 284815902739, 997741303308, 3503112067273

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The sixth row of A079216. The leftmost edge of the triangle A079222 and also its row sums shifted by one. Occurs in A073202 as row 170143779648513184865579715159814351615. Cf. A057546, A079223-A079226.

Maple
```
A079227 := n -> A079216bi(n,6);
```

a(n) = A079216(n, 6)

A079442 Number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071663.

Original entry on oeis.org

1, 1, 0, 3, 0, 9, 0, 21, 0, 45, 0, 99, 0, 195, 0, 399, 0, 801, 0

Offset: 0

Antti Karttunen, Jan 27 2003

R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Antti Karttunen, C-program for counting the initial terms of this sequence
Antti Karttunen, Illustration of initial terms for trees of sizes n=3..11

Cf. A057505, A071663, A079438, A079441, A079443, A079444.

Occurs in A073202 as row 176609070820803.

For all n >= 0, a(2n+3)/3 = A079444(n).

Name corrected by Antti Karttunen, Dec 13 2017

A073192 Number of general plane trees whose n-th subtree from the left is equal to the n-th subtree from the right, for all its subtrees (i.e., are palindromic in the shallow sense).

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 54, 155, 500, 1614, 5456, 18630, 64960, 228740, 814914, 2926323, 10589916, 38561814, 141219432, 519711666, 1921142832, 7129756188, 26555149404, 99228108222, 371886574632, 1397548389644, 5265131346368

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

The Catalan bijection A057508 fixes only these kinds of trees, so this occurs in the table A073202 as row 168.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Occurs for first time in A073202 as row 168.

Cf. also A073190.

A073192 := proc(n) local d; add( (`mod`((n-d+1),2))*Cat((n-d)/2)*(`if`((0=d),1,Cat(d-1))), d=0..n); end;
Cat := n -> binomial(2*n,n)/(n+1);

a[n_] := Sum[Mod[n - k + 1, 2]*CatalanNumber[(n - k)/2]*If[k == 0, 1, CatalanNumber[k - 1]], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2016 *)

Gat(n) = if (n == -1, 1, binomial(2*n,n)/(n+1));
a(n) = sum(i=0, n, if (!((n-i)%2), Gat((n-i)/2)*Gat(i-1))); \\ Michel Marcus, May 30 2018

a(n) = Sum_{i=0..n, (n-i) is even} Gat((n-i)/2)*Gat(i-1), where Gat(-1) = 1 and otherwise like A000108(n).

A073193(n) = (A000108(n) + A073192(n))/2.

A073203 Array of maximum cycle length sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 2, 2, 6, 2, 2, 1, 1, 2, 2, 2, 8, 2, 3, 2, 1, 1, 2, 2, 2, 10, 2, 6, 4, 1, 1, 1, 2, 2, 2, 12, 2, 8, 8, 1, 2, 1, 1, 2, 2, 2, 14, 2, 10, 16, 1, 4, 1, 1, 1, 2, 2, 2, 16, 2, 12, 32, 1, 8, 2, 2, 1, 1

Offset: 0

Antti Karttunen, Jun 25 2002

Each row of this table gives the longest cycle/orbit produced by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

A. Karttunen, Gatomorphisms (With the complete source and explanation)

Cf. also A073201, A073202, A073204.
Few EIS-sequences which occur in this table. Only the first known occurrence(s) given:.
Rows 6 and 8: A011782, Row 7: A000012, Row 12, 14: A000793 (shifted right and prepended with 1), Row 261: A057543, Row 2614: A057545, Rows 2618, 17517: A057544.

A073268 Number of plane binary trees whose right (or respectively: left) subtree is a unique "complete" tree of (2^m)-1 nodes with all the leaf-nodes at the same depth m and the left (or respectively: right) subtree is any plane binary tree of size n - 2^m + 1.

Original entry on oeis.org

1, 1, 2, 3, 8, 20, 58, 179, 576, 1902, 6426, 22092, 77026, 271702, 967840, 3476555, 12578728, 45800278, 167693698, 617037126, 2280467586, 8461771342, 31510700712, 117725789124, 441141656810, 1657559677646, 6243810767912

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

The Catalan bijection A073286 fixes only these kinds of trees, so this occurs in A073202 as row 41.

Occurs for first time in A073202 as row 41.

A073268 := proc(n) local i; if(0=n) then 1 else add(Cat(n-2^i),i=0..floor(evalf(log[2](n)))); fi; end;
Cat := n -> binomial(2*n,n)/(n+1);

a[0] = 1; a[n_] := Sum[CatalanNumber[n - 2^i], {i, 0, Log[2, n]}]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2016 *)

N=66; x='x+O('x^N); lg=ceil(log(N)/log(2));
C(x)=(1-sqrt(1-4*x))/(2*x);
gf=1+sum(k=0, lg, x^(2^k)*C(x) );
Vec(gf)
/* Joerg Arndt, Jul 02 2012 */

a(0)=1, a(n) = Sum_{i=0..floor(log_2(n))} Cat(n-(2^i))

G.f.: 1 + Sum_{k>=0} x^(2^k)*C(x) where C(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108). [Joerg Arndt, Jul 02 2012]

A079226 Number of Catalan objects fixed by five-fold application of the Catalan bijections A057511/A057512 (deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 15, 36, 108, 301, 814, 2080, 5223, 12919, 32557, 83943, 222591, 600252, 1632814, 4440240, 12043224, 32572225, 88081208, 238722759, 649725756, 1776546687, 4877740703, 13432630929, 37063472432, 102389547753

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The fifth row of A079216. The leftmost edge of the triangle A079221 and also its row sums shifted by one. Occurs in A073202 as row 9259542121261050623. Cf. A057546, A079223-A079227.

Maple
```
A079226 := n -> A079216bi(n,5);
```

a(n) = A079216(n, 5)

A079224 Number of Catalan objects fixed by three-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 43, 104, 273, 702, 1870, 4985, 13562, 37038, 102266, 283774, 793189, 2227115, 6286044, 17811751, 50672898, 144639235, 414181050, 1189365940, 3424477813, 9883578364, 28589660227, 82870288432, 240672107114

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The third row of A079216. The leftmost edge of the triangle A079219 and also its row sums shifted by one. Occurs in A073202 as row 43639. Cf. A057546, A079223-A079227.

Maple
```
A079224 := n -> A079216bi(n,3);
```

a(n) = A079216(n, 3)

A079225 Number of Catalan objects fixed by four-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 5, 11, 30, 82, 233, 680, 2033, 6164, 18923, 58768, 184045, 581105, 1846906, 5905364, 18980465, 61292929, 198758704, 646974285, 2113163707, 6923642271, 22749608810, 74946337830, 247499313730, 819154110660, 2716779932308

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The fourth row of A079216. The leftmost edge of the triangle A079220 and also its row sums shifted by one. Occurs in A073202 as row 2290625151. Cf. A057546, A079223-A079227.

Maple
```
A079225 := n -> A079216bi(n,4);
```

a(n) = A079216(n, 4)

cp's OEIS Frontend

A079227 Number of Catalan objects fixed by six-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

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A079442 Number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071663.

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A073192 Number of general plane trees whose n-th subtree from the left is equal to the n-th subtree from the right, for all its subtrees (i.e., are palindromic in the shallow sense).

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A073203 Array of maximum cycle length sequences for the table A073200.

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A073268 Number of plane binary trees whose right (or respectively: left) subtree is a unique "complete" tree of (2^m)-1 nodes with all the leaf-nodes at the same depth m and the left (or respectively: right) subtree is any plane binary tree of size n - 2^m + 1.

Views

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Crossrefs

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Mathematica

PARI

Formula

A079226 Number of Catalan objects fixed by five-fold application of the Catalan bijections A057511/A057512 (deep rotation of general parenthesizations/plane trees).

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A079224 Number of Catalan objects fixed by three-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

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A079225 Number of Catalan objects fixed by four-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

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