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.

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A089411 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A074683/A074684.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 4, 11, 9, 6, 8, 14, 14, 12, 14, 19, 17, 16, 24, 26, 30
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes. Does the non-monotone behavior continue indefinitely?

Links

Crossrefs

Cf. A089413, A089417.

A089847 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089851/A089853.

Original entry on oeis.org

1, 1, 2, 3, 8, 24, 72, 231, 764, 2574, 8840, 30796, 108528, 386308, 1386756, 5014575, 18249120, 66786738, 245642760, 907513530, 3366215688, 12531502620, 46805032560, 175341580596, 658674458208, 2480584898700, 9363748625972
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

Links

Programs

Formula

a(n) = (1/3 * A000108(n)) + (2/3 * A089848(n))

A089849 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A069772.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 20, 5, 70, 14, 252, 42, 924, 132, 3432, 429, 12870, 1430, 48620, 4862, 184756, 16796, 705432, 58786, 2704156, 208012, 10400600, 742900, 40116600, 2674440, 155117520, 9694845, 601080390, 35357670, 2333606220, 129644790
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

The number of n-node binary trees fixed by the corresponding automorphism(s). A000984 interleaved with A000108.

Links

Crossrefs

Cf. A014137, A014138, A069772, A089880.
Cf. A000984 interleaved with A000108.

Programs

  • Magma
    A089849:= func< n | n eq 0 select 1 else ((1+(-1)^n)*Binomial(n,Floor(n/2))+(1-(-1)^n)*Catalan(Floor((n-1)/2)))/2 >;
[A089849(n): n in [0..50]]; // G. C. Greubel, Feb 22 2025
  • Mathematica
    a[n_] := If[EvenQ[n], Binomial[n, n/2], CatalanNumber[(n-1)/2]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 09 2023 *)
  • SageMath
    def A089849(n): return binomial(n, n//2) if n%2==0 else catalan_number((n-1)//2)
print([A089849(n) for n in range(51)]) # G. C. Greubel, Feb 22 2025
  • Scheme
    (define (A089849 n) (if (even? n) (A000984 (/ n 2)) (A000108 (/ (- n 1) 2))))

Formula

a(2n) = A000984(n), a(2n+1) = A000108(n).
a(n) = Sum_{k=0..floor(n/2)} binomial(k+1,n-k)*Catalan(k). - Paul Barry, Feb 23 2005
From Paul Barry, Jan 23 2006: (Start)
a(n+1) = Jacobi_P(n, 2, 0, 0)*2^n*(cos(Pi*n/2)+sin(Pi*n/2)).
a(n+1) = (Sum_{k=0..n} C(n,k)*C(n+2,k)*(-1)^k)*(cos(Pi*n/2)+sin(Pi*n/2)). (End)
From Sergei N. Gladkovskii, Dec 18 2012 (Start)
E.g.f.: 1 + integral(G(0)) dx where G(k) = 1 + 2*x/(1 - 2*x/(2*x + (2*k+2)*(2*k+4)/G(k+1) )); (continued fraction).
E.g.f.: 1 + x*G(0) where G(k) = 1 + x*(2*k+1)/(k+1 - x*(k+1)/(x + (k+2)*(2*k+3)/G(k+1) )); (continued fraction).
E.g.f.: E(x) = integral( (1/x + 2)*BesselI(1,2*x) ) dx. (End)
G.f.: G(0), where G(k) = 1 + x/(k+1 - (k+1)*(4*k+2)*x/((4*k+2)*x + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 19 2013
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=0} 1/a(n) = 10/3 + 2*Pi/(3*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 2/3 + 2*Pi/(9*sqrt(3)). (End)

A127389 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A127377/A127378 and A127387.

Original entry on oeis.org

1, 1, 0, 1, 2, 4, 10, 23, 56, 138, 344, 870, 2220, 5716, 14828, 38717, 101682, 268416, 711810, 1895432, 5066030, 13586082, 36547534, 98593064, 266661162, 722953814, 1964358938, 5348367006, 14589803090, 39870312218, 109136843138
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Crossrefs

This is INVERTi transform of A086625 (appropriately shifted). I.e. INVERT([1, 1, 0, 1, 2, 4, 10, 23, 56, 138, 344, 870, 2220, 5716]) gives: 1, 2, 3, 6, 12, 26, 59, 138, 332, 814, 2028, 5118, ... (beginning of A086625)

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=(1 + x*A^2)*(1+x)/(1+x+2*x^2));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = (1 + x*A(x)^2)*(1+x)/(1+x+2*x^2).

Extensions

Generating function, PARI-program and most of the terms supplied by Paul D. Hanna, Jan 15 2007

A079440 Number of transpositions (2-cycles) in range [A014137(n-1)..A014138(n-1)] of permutation A057505 (= Donaghey's automorphism M).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 6, 6, 7, 8, 8, 9, 9, 11, 12, 12, 12, 14, 14, 14, 15, 17, 17, 18, 18, 19, 20, 20, 20, 23, 23, 23, 24, 25, 25, 26, 26, 28, 29, 29, 29, 31, 31, 31, 32, 34, 34, 35, 35, 36, 37, 37, 37, 40, 40, 40, 41, 42, 42, 43, 43, 45, 46, 46, 46, 48, 48, 48
Offset: 0

Views

Author

Antti Karttunen, Jan 27 2003

Keywords

Links

Crossrefs

From n>= 2 onward a(n) = A079438(n)/2 (with the same reservation). Cf. A079444.

Programs

  • Maple
    A079440 := n -> floor((n+1)/3) + `if`((n>=14),floor((n-10)/4)+floor((n-14)/8),0);

A079444 Number of 3-cycles in range [A014137(2n+2)..A014138(2n+2)] of permutation A057505 (= Donaghey's automorphism M).

Original entry on oeis.org

1, 3, 7, 15, 33, 65, 133, 267
Offset: 0

Views

Author

Antti Karttunen, Jan 27 2003

Keywords

Links

Crossrefs

Cf. A079440, A079442.

Formula

a(n) = A079442(2n+3)/3.

A081160 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506, with two fixed-points of A057164.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 10, 16, 35, 60, 126, 225, 462, 848, 1716, 3188, 6435, 12095, 24310
Offset: 0

Views

Author

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Keywords

Crossrefs

Bisections: A001700 (shifted right and prepended with zero) and A081161.
Cf. A081150, A081157, A081162.
Cf. A014137, A014138, A057505, A057506, A057164.

A081162 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506, with two fixed-points of A057163.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 15, 0, 56, 0, 185, 0, 655, 0
Offset: 0

Views

Author

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Keywords

Comments

Because A057163 can fix only binary trees of odd size, the even-indexed terms are all 0.

Crossrefs

Bisection: A081163. Cf. A081150, A081157, A081160.
Cf. A014137, A014138, A057505, A057506, A057163.

A089423 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A082335/A082336 (and also of A082349/A082350, to be proved).

Original entry on oeis.org

1, 1, 2, 6, 12, 120, 120, 840, 840, 5040, 5040, 55440, 55440, 720720, 720720, 720720, 720720, 24504480, 24504480, 465585120, 465585120
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Links

Crossrefs

Bisection: A089431.

A126295 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A125987/A125988.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 5, 4, 4, 4, 4, 4, 7, 8, 7, 6, 10
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Crossrefs

Fixed points themselves are given in A126298/A126299.
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