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.

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))

A079217 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 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, 0, 1, 10, 0, 0, 0, 0, 0, 1, 11, 5, 0, 1, 0, 0, 0, 1, 18, 0, 2, 0, 0, 0, 0, 0, 1, 21, 11, 0, 0, 1, 0, 0, 0, 0, 1, 34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 35, 26, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 68, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 69, 66, 0, 0, 0, 0, 1, 0, 0
Offset: 0

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Links

Crossrefs

The row sums equal to the left edge shifted left once = A057546 = first row of A079216 (the latter gives the Maple procedure PFixedByA057511).
Cf. A079218, A079219, A079220, A079221, A079222 and A003056 and A002262.

Programs

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen Jan 03 2002

Keywords

Crossrefs

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.

Programs

Formula

a(n) = A079216(n, 6)

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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