cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A057546 Number of Catalan objects of size n fixed by Catalan Automorphism A057511/A057512 (deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 18, 21, 34, 35, 68, 69, 137, 148, 316, 317, 759, 760, 1869, 1915, 4833, 4834, 12796, 12802, 34108, 34384, 92792, 92793, 254752, 254753, 703083, 704956, 1958210, 1958231, 5485330, 5485331, 15427026, 15440591, 43618394, 43618395, 123807695, 123807696, 352561832, 352664217, 1007481494, 1007481495, 2887387009
Offset: 0

Views

Author

Antti Karttunen, Sep 07 2000

Keywords

Comments

Greater than A003238 because there exists also parenthesizations like ((() (())) ((()) ())) and (((()) ()) (() (()))) which are fixed by recursive deep rotation, corresponding to Catalan mountain ranges below:
...../\..../\............................./\......../\
../\/\../\/\.....and.its."dual"....../\/\../\/\
./____\/____\......................./____\/____\
It's obvious that a(p) = a(p-1)+1 for all primes p.

Links

Crossrefs

The first row of A079216. The leftmost edge of the triangle A079217 and also its row sums shifted by one. Occurs for first time in A073202 as row 12. Cf. A057513, A079223-A079227, A034731, A003238.

Programs

  • Maple
    with(numtheory,divisors); A057546 := proc(n) local d; if(0=n) then RETURN(1); else RETURN(add(A079216bi(d-1,n/d),d=divisors(n))); fi; end;

Formula

a(0)=1, a(n) = A079216(n, 1) = Sum_{d|n} A079216(d-1, n/d). - Antti Karttunen, Jan 03 2003

A079216 Square array A(n>=0,k>=1) (listed antidiagonally: A(0,1)=1, A(1,1)=1, A(0,2)=1, A(2,1)=2, A(1,2)=1, A(0,3)=1, A(3,1)=3, ...) giving the number of n-edge general plane trees fixed by k-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 5, 2, 1, 1, 6, 11, 3, 2, 1, 1, 10, 26, 8, 5, 2, 1, 1, 11, 66, 18, 11, 3, 2, 1, 1, 18, 161, 43, 30, 5, 5, 2, 1, 1, 21, 420, 104, 82, 6, 14, 3, 2, 1, 1, 34, 1093, 273, 233, 15, 38, 5, 5, 2, 1, 1, 35, 2916, 702, 680, 36, 111, 6, 11, 3, 2, 1, 1, 68, 7819, 1870
Offset: 0

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Comments

Note: the counts given here are inclusive, e.g. A(n,6) includes the counts A(n,3) and A(n,2) which in turn both include A(n,1).

Links

Crossrefs

A(n, A003418(n)) = A000108(n). The first row: A057546, second: A079223, third: A079224, fourth: A079225, fifth: A079226, sixth: A079227. Cf. also A079217-A079222.

Programs

  • Maple
    with(combinat, composition); # composition(n,k) gives ordered partitions of integer n into k parts.
[seq(A079216(n),n=0..119)]; A079216 := n -> A079216bi(A025581(n), A002262(n)+1);
A079216bi := proc(n,k) option remember; local r; if(0 = n) then RETURN(1); else RETURN(add(PFixedByA057511(n,k,r),r=1..n)); fi; end;
PFixedByA057511 := proc(n,k,r) option remember; local ncycles, cyclen, i, c; ncycles := igcd(r,k); cyclen := r/ncycles; if(0 <> (n mod cyclen)) then RETURN(0); else add(mul(A079216bi(i-1,ilcm(r,k)),i=c),c=composition(n/cyclen,ncycles)); fi; end;

Formula

A(0, k) = 1. A(n, k) = Sum_{r=1..n where r/gcd(r, k) divides n} Sum_{c as each composition of n/(r/gcd(r, k)) into gcd(r, k) parts} Product_{i as each composant of c} A(i-1, lcm(r, k))

A079222 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the six-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 14, 14, 9, 0, 1, 38, 42, 28, 2, 0, 1, 111, 124, 90, 0, 0, 6, 1, 332, 379, 285, 5, 0, 27, 0, 1, 1029, 1178, 914, 0, 0, 110, 0, 0, 1, 3232, 3742, 2955, 14, 1, 429, 0, 0, 0, 1, 10374, 12024, 9666, 0, 0, 1614, 0, 0, 0, 0, 1, 33679, 39200, 31853, 42, 0
Offset: 0

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Comments

Note: the counts given here are inclusive, i.e. T(n,d) includes also the counts A079218(n,d) and A079219(n,d).

Crossrefs

The row sums equal to the left edge shifted left once = A079227 = sixth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079221 and A003056 & A002262.

Programs

A079218 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the two-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 0, 1, 11, 14, 0, 0, 1, 26, 36, 1, 2, 0, 1, 66, 94, 0, 0, 0, 0, 1, 161, 253, 0, 5, 0, 0, 0, 1, 420, 668, 2, 0, 0, 2, 0, 0, 1, 1093, 1807, 0, 14, 1, 0, 0, 0, 0, 1, 2916, 4902, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7819, 13436, 5, 36, 0, 5, 0, 2, 0, 0, 0, 1, 21304, 37016, 0, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Comments

Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d).

Crossrefs

The row sums equal to the left edge shifted left once = A079223 = second row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 & A002262.

Programs

A079219 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the three-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 3, 1, 8, 0, 9, 0, 1, 18, 2, 22, 0, 0, 1, 43, 0, 60, 0, 0, 0, 1, 104, 5, 159, 1, 0, 3, 0, 1, 273, 0, 428, 0, 0, 0, 0, 0, 1, 702, 14, 1143, 0, 1, 9, 0, 0, 0, 1, 1870, 0, 3114, 0, 0, 0, 0, 0, 0, 0, 1, 4985, 38, 8505, 2, 0, 28, 0, 0, 3, 0, 0, 1, 13562, 0, 23475, 0, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Comments

Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d).

Crossrefs

The row sums equal to the left edge shifted left once = A079224 = third row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 and A002262.

Programs

A079221 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the five-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 0, 0, 0, 1, 6, 2, 1, 0, 5, 1, 15, 0, 0, 0, 20, 0, 1, 36, 5, 0, 1, 65, 0, 0, 1, 108, 0, 2, 0, 190, 0, 0, 0, 1, 301, 11, 0, 0, 501, 0, 0, 0, 0, 1, 814, 0, 0, 0, 1265, 0, 0, 0, 0, 0, 1, 2080, 26, 3, 2, 3105, 1, 0, 0, 0, 5, 0, 1, 5223, 0, 0, 0, 7695, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Comments

Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d).

Crossrefs

The row sums equal to the left edge shifted left once = A079226 = fifth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 and A002262.

Programs

A079220 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the four-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 0, 1, 11, 14, 0, 4, 1, 30, 36, 1, 14, 0, 1, 82, 102, 0, 48, 0, 0, 1, 233, 293, 0, 153, 0, 0, 0, 1, 680, 860, 2, 488, 0, 2, 0, 0, 1, 2033, 2575, 0, 1550, 1, 0, 0, 4, 0, 1, 6164, 7838, 0, 4920, 0, 0, 0, 0, 0, 0, 1, 18923, 24148, 5, 15672, 0, 5, 0, 14, 0, 0, 0, 1
Offset: 0

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Comments

Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079218(n,d).

Crossrefs

The row sums equal to the left edge shifted left once = A079225 = fourth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 & A002262.

Programs

Showing 1-7 of 7 results.

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